"Now the spirits come!! Come, spirits of power, come spirits of sight! Show to me the awful things occurring here tonight!"
—Madame Clairvoya, Luigi's Mansion
Ah, spirits. Perhaps no other fantastic beings in all of fictiondom can be as vague as to what they are than them.
There are several questions you have to ask, though, if you wish to have them in your story:
Are they immaterial/incorporeal, and if so, to what degree?
Most are immaterial, but they can take material form if they choose to.
In Bleach, ordinary human spirits are called "Pluses," usually created when someone dies with Unfinished Business. Pluses tend to stick near places or people that were important to them in life (tethered to the location by a long chain), and sometimes might even realize that they're dead. However, since they're Invisible to Normals, they slowly grow resentful and insane through isolation and/or neglect. When this begins, they turn into spirits called "demi-Hollows", which are still normal Pluses with a hole slowly growing on the spot where their chain is connected, symbolizing their decaying connection to their humanity. When the chain has eroded all the way (or when external factors prompt a change), they become full Hollow with an unquenchable thirst for souls. If a shinigami sends the soul off to Soul Society (the afterlife), they'll be safe from transforming into a hollow, but will now begin a second life living in a poor, destitute city based roughly on feudal Japan. The lucky few qualify to become shinigami themselves (although, given the average life expectancy of a shinigami, how "lucky" this is can be disputed). The shinigami and Hollow also have their own hierarchy and ranking, and despite being the "good guys", the shinigami are essentially nobility and treat normal Pluses at best with indifference and at worst with contempt.
Date A Live's Spirits are beings which are occasionally sucked into this world, causing "spacequakes" (a phenomenon that destroys everything within a spherical area of effect) in the process. And they happen to be top class Bishojo as well. To stop them from wreaking havoc on this world by accident, without killing them? The main character Shido has to date them.
The high spirits of Adam R. Brown's Astral Dawn series are very different from the average spook found haunting houses. The high spirits were once mortal humans before they died and ascended to Averya (Heaven) or Nazyra (Hell). They evolved to become so powerful, they gained total dominion over their created worlds and even revisited Earth to become the gods and legends of ancient mythologies.
In The Dresden Files, Humans are made of three things: Body, Spirit, and Soul. Ghosts and some creatures from the Nevernever are made of only Spirits and other creatures from the Nevernever (such as Fae) are made of both Body and Spirit. Finally, some very special kinds of ghosts can have both Spirit and Soul.
Beings from the Nevernever can be summoned or cross over on their own. For those without bodies they can create material bodies for themselves made of ectoplasm, which dissipates without a trace if they are killed or leave (or no longer have enough magic to maintain the body).
Certain kinds of spirits (ghosts, demons) can possess humans and animals (e.g. cats) but not all of them, apparently. Permanent bonds are possible via mutual pledges of loyalty.
All spirits seem ageless, but they're not invulnerable.
Spirits always stay the way they are.
They can be consumed/deformed by other beings, usually by bigger and meaner spirits, though mortal wizards can feed on them, too.
Since spirits are incapable of evolution, their True Names never change, and whoever knows it can force them to do their bidding.
The Stormlight Archive has the spren, which are associated with various phenomena. Whether they're drawn to said phenomena, created by them, or cause them is unknown both in and out of universe. These phenomenon include wind, rain and fire, natural processes like rot, emotions like fear and passion, and even more esoteric things like music, creation, and honour. Some spren can move small objects, and they appear to be heavily connected with various forms of magical powers.
In Lord of the Rings and the rest of Tolkien's verse (The Silmarillion, The Hobbit, The Children of Hśrin), there are basically two kinds of spirits. The first, Ainur (Valar and Maiar) are the local version of angels and demons (ie. Valar and Maiar begin as angels, but can become corrupted and turn into demons); they exist simulataneously in two worlds (wraith-world and the real world), can change shape and become immaterial at will (unless they are in Mode Lock like Gandalf and Sauron), are immortal and happy about it, come from Valinor and are the source of most Functional Magic in the verse. The second type (Nazgūl, Wraiths, ghosts etc) are spirits who were once souls of Men (or Elves), they live mostly in the wraith-world and have problems interacting with the real world, are usually immaterial, immortal but remarkably not happy about their condition and have less powerful magical abilities; they are also likely to be evil, because souls of good people obediently go to the halls of Mandos to reach the afterlife (though the Nazgūl, at least, are actually victims of The Corruption).
Religion...as in all of them. Trinitarian Christianity has the Holy Ghost, Buddhism has Hungry Ghosts (although, they're more like ghouls than conventional spirits), various religions and mythologies have regular actual ghosts, and of course Shinto and Animism have everyone. And everything.
Suffice it to say, just about everything in mythology that wasn't human was either a spirit or descended from one.
More specifically, nWoD spirits are incorporeal beings associated with concepts, Anthropomorphic Personification but usually without the anthropomorphic part. They sustain themselves by hanging around places associated with their concept; river spirits with rivers, grief spirits with funeral parlors, spirits of mugging with back alleys, etc. They can also try to encourage or nurture the concept that feeds them, which is bad news when dealing with a spirit of murder or something similar.
Both worlds also give spirits a hierarchical structure, organizing them into courts with various ranks, but with different twists. In nWoD, each spirit is technically an independent actor that shares a Choir with like types - for instance, each dove spirit gets along with other dove spirits and is a member of the Choir of Birds. In cWoD, each spirit is effectively a "representative" of a greater totem, which may itself fall in with the brood of a totem more powerful than itself - for instance, each dove spirit is actually an aspect/representative of Dove, who is herself allied with the brood of Unicorn, and so forth.
In the Dragon Age series, spirits and demons are the same thing. They inhabit the Dream World called the Fade but can be summoned into the material world (or enter it on their own in certain locations). They seem to be ageless but can be killed, whether in the material world or in the Fade. Both demons and benevolent spirits can possess humans, though the former tend to turn them into Humanoid Abominations. Each demon is associated with one of five sins (corresponding to the negative emotion it feeds upon); benevolent spirits seem to be associated with virtues but because encounters with them are so rare, no established classification exists.
The Legend of Zelda calls its spirits Poes. They take the form of ghostly specters holding lanterns, and there are several quests that involve collecting them.
In Shining the Holy Ark three good spirits are in charge of magical floating spaceship, the titular Ark. However the Ark gets destroyed so they escape in a pod, crash onto our heroes and possess them. This heals our three main heroes (as they were squashed by the escape pod) and makes sure they can never die, gives them additional powers and allows themselves to heal up while they inhabit their bodies. Meanwhile an evil spirit possesses the King and his advisers in an attempt to revive the 1000 year kingdom and bring fourth an age of darkness.
In Avatar: The Last Airbender, the spirits inhabit a Spirit World, but many of them have the ability to take form in the 'real' world, as well. In terms of vulnerability, it varies from spirit to spirit. On the whole, taking form in the real world leaves them vulnerable to attacks, and if they take on a permanent living body, such as the moon and ocean spirits, or the eponymous avatar, they can be permanently killed. Powers also vary: some are a Anthropomorphic Personifications and have abilities linked to that, while others are random assortments. As far as genies, fairies, souls and ghosts are concerned, its kept ambiguous or not really mentioned.