Usually acquired by spending skill points, occasionally otherwise
Higher skill levels may cost more than the lower ones
More advanced perks may cost more than the basic ones
Skill scores and perks often coexist side-by-side, mixed-and-matched in a myriad of ways, and many games (especially contemporary RPGs) go out of their way to blur the line between the two.
See also The Six Stats, Skill Point Reset, No Stat Atrophy.
Tabletop game examples:
GURPS is a classic example. Character traits fall fundamentally into four distinct groups: attributes (the four base stats), skills (based on said attributes and then improved individually), advantages (perks) and disadvantages (essentially 'anti-perks', handicaps to saddle one's character with for more character depth and bonus points...okay, primarily the bonus points).
Dungeons & Dragons was late to this particular party for once. The now-familiar skills-and-feats scheme was introduced only with the game's third edition; before then, support for skills for non-thief characters was noticeably delegated to optional rules or even altogether absent, and "perks" existed almost solely in the form of racial and class abilities that might get unlocked with advancing level but generally didn't involve much choice. Moreover, both pre- and post-third edition these skills were and are handled more like perks themselves, with characters able to choose to have a given skill or not but little in the way of means to improve it further after that.
Both the Storyteller and the Storytelling System are based around "traits", which are basically a hybrid mash-up of stats ("attributes"), skill scores ("abilities" in oWoD, "skills" in nWoD), and upgradable perks ("advantages"). The latter include both storyline perks (like background) and active abilities (like the vampiric Disciplines). Attribute scores range from 1 to 5, abilities/skills and advantages from 0 to 5. Occasionally, levels up to 10 may be allowed. Trait levels are acquired by spending character points.
The Serenity Role Playing Game has a nontraditional version of this. Skills and perks are purchased from a shared point pool at character creation and changeable at GM discretion (the RPG does not use a Class and Level System). There's a set of core skills that can be used with dice up to d6, then specialized into sub-skills that can use from d8 to d12+d2. Perks, called traits, are broken down into assets and complications and provide various bonuses and penalties.
Ironclaw has skills and Gifts, adding skill marks increasing the size of the dice rolled. The biggest changes between editions included the consolidation of several skills and changing many specific skills to Gifts, as well as instituting flat costs of 4 XP per skill mark and 10 XP per Gift rather than varying costs, and making spells Gifts.
Video game examples:
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Diablo II features a hybrid skill score/perk system, wherein each class has three unique skill trees consisting of several tiers of perks. Individual tiers are unlocked one by one at certain character levels, after which any number of skill points (gained at each level and from some quests) can be invested into any unlocked perk, increasing its efficiency and often giving bonuses to more advanced perks derived from it. On a side note, Diablo II has one of the earliest implementations of the aura-type perks (with its Paladin class).
Mass Effect 1 has a hybrid skill score/perk system wherein "talents" encompass both passive (armor, weapon) and active (biotic attacks) skills. Talents can be leveled up multiple times, increasing their efficiency and unlocking additional perks and even further upgradeable talents at certain levels (forming an implicit skill tree). Mass Effect 2 and Mass Effect 3 replace them with "powers"—active combat skills, which are essentially perks with up to five (mutually exclusive at level 4+) upgrades available for each of them. Most talents and powers depend on the character's class and are available from the start, but some are only unlocked after certain story events.
The original Deus Ex features both skill scores and perks (dubbed "augmentations"). Skill scores are upgraded using skill points acquired from completing missions, have four levels (Untrained, Trained, Advanced, Master), and improve your performance with different types of weapons, or non-combat actions (lock-picking, electronics, medicine, etc.). Augmentations are picked up as items on missions and implanted into Denton's body slots, giving him new abilities. They can likewise be upgraded. Deus Ex: Human Revolution does away with skill scores and instead translates experience points into Praxis Points, which in turn can be invested into any augmentation unlocked in the perk tree (justified by that all augs are already built into Jensen and only need to be activated).
Path of Exile has an enormous perk tree consisting of 1300 nodes available to each class, although all nodes are passive bonuses (active skills are instead imbued into items). Furthermore, a majority of nodes are unspectacular permanent bonuses to one of the character attributes (strength, dexterity, intelligence), but buying them is required to get to the juicy, named perks that provide significant combat advantages.
RPG — Eastern
Uncharted Waters: New Horizons features a combination of seven skill scores (ranging from Leadership, through Knowledge, to Swordsmanship) which increase your performance as a fleet captain, and five perks: Celestial Navigation lets you auto-sail to any known port, Cartography allows you to sell maps, Gunnery drastically improves combat performance, Accounting tells you best places to sell your goods, and Negotiation lets you haggle for better prices.
Final Fantasy X is famous for the Sphere Grid and its potential flexibility, with nodes that either increase certain stats or provide a new active ability. Each node requires the appropriate sphere to activate it and certain areas of the grid are separated by locked nodes of various levels. The original version have a somewhat linear progression path for each character(with the exception of Kimahri, who starts in the center with access to almost every other character-specific grid area) giving them a more pre-determined role early on, but can access other parts once they complete their paths or unlock the lock nodes separating each area. The International version adds a second grid layout, Expert, in which every character starts at the center of the grid and are able to go anywhere they wish, though overdrives and weapons remains unique to each character.
The Crystarium system Final Fantasy XIII consists of class-and-character-specific (mostly linear with minor sideways branching) perk trees, whose nodes have increasing "crystogen point" costs. Most nodes provide bonuses to either Strength, Magic, or Health, but some unlock class/role-specific moves. In Final Fantasy XIII-2, each character has only one linear "perk tree", but its nodes are technically empty slots where the player can place perks (bonuses or abilities) of any of the six available classes (up to 99 per class) by investing crystogen points.
RPG — MMO
Star Wars: The Old Republic features a two-fold upgradable perk system, split into class-specific Abilities and Prestige Class-specific Talents. Abilities are mostly active powers that can be purchased from or upgraded by class trainers for credits after reaching certain levels. Talents are mostly passive bonuses with some unique active powers thrown in that form three loosely connected specialization-specific trees. Each level beyond 10 gives you a Talent point to buy a new talent or upgrade an old one. In addition, there are the Crew Skills, which allow crafting and resource gathering and are improved by using them, though you are limited to three at any time.
Skills and traits are selected by the player and provide mainly passive bonuses in combat. Your skills increase the potency of your ship and away team powers (for example, Starship Graviton Generators improves gravity-related bridge officer powers such as Tractor Beam and Gravity Well). Attributes apply only to the character who has them. Individual species have inherent attributes (e.g. Bajorans get a bonus to HP heals on the ground), and player characters can select from a pool of additional traits (bridge officers have the latter built-in).
Completing kill and damage accolades provides small passive bonuses to damage and toughness. For example, destroying 200 Borg ships and getting the "Nanoprobe Immunity" accolade gives a 2% damage boost against Borg ships, and taking 15,000 kinetic damage with your captain for the "Punching Bag" accolade gives +2 to your away team's kinetic damage resistance.
RPG — Western
The Fallout series has a skill system based around seven static attributes (ranging from 1 to 10), skill scores (measured in percent and upgraded at every level), traits (essentially perks with both advantages and disadvantages that you pick at the beginning of the game), and learnable perks (which you can take every three levels).
The underlying game system of The Elder Scrolls series is based mainly around skill scores, which are increased by using them (which also indirectly increases their governing character stats), but Oblivion adds a number perks, which are unlocked by reaching certain levels in skills: e.g. a Power Attack at Blade 25, no shield wear-out at Shield 50, jumping off water surface at Acrobatics 100, etc. Skyrim expands the available perks to an entire tree (one per skill score); perks now have prerequisite perks and corresponding skill score levels and can be bought and upgraded with perk points (gained with every level up) once unlocked.
In Dragon Age: Origins skill scores ("skills") and perks ("talents" and spells) exist in parallel, although the latter are much more important. Eight (11 in Awakening) skills can be leveled up four times, providing passive bonuses to anything from persuasiveness, through pickpocketing, to combat efficiency. Talents, meanwhile, are distinct moves, auras, passives, and upgrades grouped into "trees" of three (four in Awakening) linear four item-long branches (mages get spells instead of talents but they work the same way). There are also class- and specialization-specific perk trees. In addition to being part of a skill tree, new perks have stat and (in case of weapon talents) skill level requirements. Talent points are gained at every level, and skill points, every three (two for rogues) levels. Dragon Age II does away with the skills but expands the talent trees to branch out more.
Alpha Protocol uses a system similar to Mass Effect 1 in that your skill points both improve your proficiency directly and, at predefined points along the tree, grant access to special actions such as Bullet Time or HUD indicators as to enemies' location, disposition, and facing. Various other perks are granted for fulfilling story conditions.
Being based on the d20 System, the Knights of the Old Republic games feature both skill scores ("skills") and perks ("feats" and "powers"). Skill scores range from hacking, through persuasion, to first aid and are upgradeable multiple times per level (skill points gained depend on INT stat). General feats (mostly combat moves, but also stat and skill bonuses) usually come with one or two upgrades and can be taken every other level (depending on the class). Powers can only be learned by the Jedi at a rate of one per level (more for some classes). Powers associated with the Light or Dark Side of the Force additionally gain bonuses from your Karma Meter standing.
Pillars of Eternity has two separate skill point pools, one each for combat and non-combat abilities.
The Witcher has an expansive skill tree focusing on four primary attributes (Strength, Dexterity, Stamina, Intelligence) and talents (Signs, steel sword, silver sword). Talents have a prerequisite attribute (i.e. you cannot learn lvl.3 "strong" sword style without lvl.3 Strength), and the Signs additionally have to be unlocked at Circles of Elements first. Individual perks are leveled (bronze, silver, gold), as are skill points (silver points are earned from level 15 onwards; gold ones, from level 30). Lastly, there are some perks that cannot be learned normally but have to be unlocked by drinking unique "mutagen" potions.
The Jedi Knight series features a number of upgradeable Force Powers. In the first game and Jedi Academy, you gain skill points after every mission (first game gives bonus points for finding secret locations, while JA has optional missions to gain extra points) and can spend them on any power you want. In Jedi Outcast, on the other hand, force powers are upgraded automatically according to a fixed progression after every mission (you can still distribute points freely in the multiplayer mode, though). The powers are further subdivided into Neutral, Light, and Dark and in Academy, your mentors will have comments if you prefer one Side over the other.
In XCOM: Enemy Unknown, each soldier can select one of two perks at every rank (which is basically a disguised XP-based Character Level) except Rookie, Squaddie, and Major (only one perk is available at each of the latter two). The perks are usually split between two core functions of a class: Assault chooses between offensive and defensive perks; Sniper, between shooting more precisely and more often; Heavy, between More Dakka and more Stuff Blowing Up; and Support, between healing and other supporting tasks (smoke grenades, cover fire). Additionally, there are psychic powers, which only a few soldiers can acquire, and more powerful abilities are unlocked by using the more basic ones.
No One Lives Forever 2 includes a skill score system that represents various aspects of being a super-spy (stealth, marksmanship, gadgets, etc.). Each skill score can be upgraded four times, giving various passive bonuses to the respective basic action (hiding, shooting, breaking codes, etc.). Each level costs progressively more skill points, which are obtained from completing missions and finding manuals scattered throughout the game.
The Dating SimShira Oka: Second Chances includes a hybrid skill score/perk system as part of its Character Customization: you gain experience points during the game for succeeding at certain tasks or selecting certain choices and can use them to upgrade your stat scores or "quirks". Upgrading your stats (Intelligence, Fitness, Creativity, and Charm) enables you to build them up faster and get better grades, whereas upgrading your quirks follows a 3-tier system (you start out with universally negative quirks — Lazy, Jinxed, Scatterbrained, Amnesiac — that only hinder you and can upgrade them twice to actually positive ones) and gives you additional dialogue options and in-game benefits (e.g. upgrading Lazy to Determined decreases the probability of you showing up late to dates). Finding a balance between upgrading stats to keep your grades high enough and quirks to give you more XP opportunities is essential to getting through the game with a minimum number of Groundhog Day Loops.