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The number system used by most of the modern world today is called the decimal system, involving ten digits ("Base 10"). Sometimes, if a writer wants to portray a society as being significantly alien to our own, they will include a mention of an alternative number system for this society, with the "base" being a number other than ten.

This may be used to indicate the collective intelligence of the society that produced it, if it is portrayed as more sophisticated or more primitive than our system. There may also be an inferred correlation between the ten digits in our number system and the ten digits on the average pair of human hands. Therefore, a race of aliens with Four-Fingered Hands may use a base eight number system. Finally, it is very common for robots or other computer-based intelligences to count in base two.

This sort of thing is generally used as an insignificant throwaway joke, with "ordinary" numbers applying before and after, as it may be a difficult concept for some viewers to grasp.

Examples:

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Comicbooks

In ElfQuest, the elves use base 8 because of their Four-Fingered Hands — that page has more details.

Fan Works

Aeon Natum Engel: The narration from the Migou POV and their dialogue goes to great lengths to convey their alien thought processes, including a base-36 numbering system.
In Hoofstuck, Twilight Sparkle mentions that Equestria uses base eight rather than ten, since there's nothing on a pony you can use to count to ten (whereas four hooves times two make more sense). They used to run with base four originally, symbolising the four races (earth ponies, pegasi, unicorns, alicorns) but the earth ponies took offense at being considered the "zeroes".

Film

Behind-the-scenes material on Avatar claims that the Na'vi also use base 8 for the same reason as the ElfQuest elves.

Literature

In the Star Wars Expanded Universe, the Hutts use a base 8 system due to only having four fingers on each hand. Being Hutts, they don't always tell this to their business partners, most of whom use base 10.
In the Discworld books, trolls apparently have a "base Many" system (actually base four). As in, "one, two, three, many, many-one, many-two...
A subversion in The Restaurant at the End of the Universe: When Douglas Adams revealed that the question which produced the Ultimate Answer (42) was What do you get if you multiply six by nine?, somebody pointed out that the math actually did add up, using base 13. Adams responded, "I may be a sad individual, but I don't make jokes in base 13."
According to the appendices of The Lord of the Rings, elves habitually count in base 12.
The gukuy in Eric Flint's Mother of Demons count using an eight-base system.
It's never mentioned in the books themselves, or the series, but the "Gallifreyan numerals" used on the spines and chapter headings of the Doctor Who New Series Adventures (9th and 10th Doctors) are in base 7.
In Leo Frankowski's The Cross Time Engineer series, the new civilization Conrad Stargard starts up in Medieval Europe uses base 12 mathematics, because Stargard believes it's "more accurate" than decimal mathematics.
- Not so much more "accurate" as more useful. Bulk goods are bought and sold in dozens and grosses. 12 factors into 2, 3, 4, and 6. It is far easier to cut a cake/pie into 12 pieces than ten without involving a protractor.
In Learning the World by Ken MacLeod, the aliens are four-fingered, and count in base 8. When they learn that humans use base 10, their reaction is that having a base that isn't a power of two must be awfully inconvenient.
In A Fire Upon the Deep by Vernor Vinge, the doglike Tines have two different number systems: one where they count "by legs" (in base 4) and one where they count "by fore-claws" (in base 10). Confusion between these two systems leads to the accidental meeting of two of the major characters. Amdiranifani is housed in room 33, Jefri is supposed to be imprisoned in room 15 (33 in base 4), and the guard who's taking him there uses the wrong numbering system.
On Gor, the alien species the Kurii use base-12, presumably because they have 12 digits on their "hands."
Little Fuzzy, a series begun by H. Beam Piper, uses a modified form of base 5. 1, 2, 3, 4, one hand. At 125, they've reached a hand of hands. It then goes to many, and many many. The fuzzies soon adopt the human's base 10 system.
Larry Niven and Jerry Pournelle's The Mote in God's Eye. The Moties have a total of 12 digits on their right hands and use base 12.
The Kzinti of Larry Niven's Known Space count in base eight.
The Culture uses a nonary (base nine, that is) system for their writing, although it's actually binary in a 3x3 square. Binary is mentioned as being used by some civilisations, and powers of two are the closest thing to a universally-regarded 'round number'.
In Stranger in a Strange Land, the Martians apparently have a numerical system based on three and powers of three. "Three fulfilled," they call it...
According to The Klingon Dictionary, the Klingons used to count in a trinary (base-three) system, but have since switched over to decimal.
In Footfall, the Fithp use base 8.
In Contact, a message is encoded in base 11, by someone or something capable of messing with the values of mathematical constants.
The aliens in the Rama books appear to use base 3.
In The Iron Standard by Henry Kuttner, the six-fingered Venusians use base-12.
Out Of The Dark gives us the Shongairi, who count in base-twelve. Possibly.
Zamonien in Walter Moers' novels uses base 8. The author even created new symbols for the numbers.
In the Fighting Fantasy book Rebel Planet, the two-fingered Arcadians count in binary. Converting binary to decimal is necessary to solve a couple of puzzles, though the reader is fortunately provided with a handy grid.
Centaurians in The Pentagon War have 4 tentacle-fingers on each of 4 hands. When a Centaurian named Torra Zorra reads that a cable's diameter is 3 x 10^-5 meters, we get this parenthetical aside:
(Curse the human penchant for powers of ten! Torra always had to mentally convert their numbers to base sixteen, just to get a handle on them.)
The Megarites of the Star Trek universe use base 8, according to Star Trek: Ex Machina.
In Battlefield Earth the Psychlos use a base-11 system (they have an extra finger on one hand). While this causes the human characters endless frustration when trying to work with Psychlo mathematics, it gets nonsensical when other alien races share these complaints, lauding the values of a sensible base-10 system and concluding that the Psychlos were just being perverse when designing their math.
- It doesn't help that Psychlo mathematics requires that the user have a basic knowledge of the Psychlo homeworld's geography. Yet another bit of evidence that they're just trolling the universe.
Gully dwarves from Dragonlance, being astonishingly stupid, can only count up to two. Any higher quantity is described as "One and one and no more than two."

Live-Action TV

Stargate SG-1:
- A throwaway comment in the episode "The Fifth Race" implies that the Ancients counted in base eight.
- Also invoked when they nearly set off a Tobin mine by entering in the wrong code due to Daniel failing to factor in a zero into his translation. He argues the Phoenicians they were descended from never used a number zero, but Carter points out in order to program something as complex as a mine, the Tobin's would have had to have added a zero component.
  Carter: Trust me; its a math thing.
- There's a fan theory that the constellations on the gate actually correspond to digits in a base-38 number system. Which means that a gate address is not six arbitrary points in space with the destination at the intersection,*
  which contradicts the in-series assertion that all addresses are unique
  but three two-digit base-38 coordinates.
In The Frantics' sketch "Roman Numerals" a Roman citizen is baffled by the new decimal system.
Customer: How much is "forty-four" in real numbers?
Shopkeeper: XLIV *
pronounced "ex-ell-eye-vee"
.
Customer: Well why don't you just say XLIV? Who can remember "forty-four?"
In Star Trek: The Next Generation, the Bynars use base 2 (i.e. binary).
Babylon 5:
- The Mimbari use base eleven (a byproduct of using fingers, toes, and the head as "digits" for counting).
- According to tie-in RPG materials, the Dilgar used base 25.

Tabletop Games

In Traveller, the various alien species use different base mathematics. The Aslan use Base 8, the Hivers use Base 16, and the Droyne use Base 6. Most of the various Human Aliens, as well as the Vargr, use Base 10.
The Warhammer 40000, the Tau use base 8 owing to their Four-Fingered Hands, leading to a delicious little Stealth Pun: their battlesuits' designations are based on their size, such as XV25 for the smaller Stealth suits, XV8 for the main Crisis suit, or XV88 for the heavy Broadside variant. Which means that the new XV9 Hazard Close Support Armour units are taking things Up to Eleven.

Videogames

In the Myst games, the D'ni have a base 25 system, in keeping with the games' general tendency to use powers of five as Arc Numbers.
The aliens in Iji use a ternary number system.
During Portal's boss fight, after GLaDOS takes a missile hit she'll announce "Two plus two equals... ten! In base four I'm fine!"
There is a popular theory among Half-Life fans that the Combine use a base-17 system, based on how prominent 17 seems to be. If nothing, it reinforces the utterly alien nature of the Combine.
The Kilrathi from Wing Commander use Base 8 for their numbering system, given that they have a total of eight fingers. For the most part this isn't really mentioned much, but it's prominent in the dates for history of the Kilrathi war from their viewpoint as done in the manual for Armada, "Voices of War".
Halo fans speculate that the Forerunners might have counted using a Base-7 counting system.
A significant portion of the game Rama, based on Arthur C. Clarke's novel series of the same name, involves solving mathematical puzzles based on the native number systems of the Avians (base 16) and Octospiders (base 8). (Which is at odds with the Ramans, who use base 3.)
- Of course, both the Avians and Octospiders aren't Ramans. They're just samples of other space-faring species that the Ramans had gathered.

Webcomics

In Housepets!, a mouse named Spo came from a very large family. How large? The sibling born immediately after him was named Spp...
In Girl Genius' world of mad science there's bound to be examples showing up: "Tell the landlord that he is to stop trying to make change in base eight, or he'll pay his taxes in base twelve."
Cowbirds in Love points out that every base is base 10 (from the perspective of anyone using it). For bonus points, the alien in the comic uses base 4 and appears to be four-fingered.

Web Originals

In some of the Chakona Space stories, Caitians are mentioned to be using base 8 math and the less mathematically minded ones struggle with everyone else's usage of base 10 math.

Western Animation

In Futurama, robots sometimes use base 2.
The Schoolhouse Rock music video for "Little Twelvetoes" briefly touches on the idea of what counting with a base-twelve system would be like, and demonstrates with the titular twelve-fingered alien character.
In The Simpsons Principal Skinner wishes he had enough funding for his school to buy, amongst other things, "math books that don't have that base 6 crap in them". This is almost certainly a reference to New Math, a way of teaching mathematics that was briefly popular in The Sixties but very rapidly fell out of favour.

Real Life

Computers work in Base 2 because the only input signals they can distinguish between are "on" and "off". Each one is called a "bit". The de facto standard of a byte establishes it as 8 bits, prompting people familiar with computer science to use the hexadecimal system (base 16) to represent a byte of information in two digits.
- Bases that are powers of 2 are fairly popular in computer science, including the above-mentioned hexadecimal (base 16) system and octal (base 8) which is most commonly used with the UNIX chmod command for setting file permissions
"New math" had learning to do operations in bases other than 10 as part of its syllabus. It was also one of the things it was mocked for.
Ancient Maya used a base 20 system.
Ancient Babylonians counted in base 60. This is reflected in the modern measurement of time (hours, minutes and seconds), as well as angular measure (degrees, minutes, seconds).
Vestiges of base 20 remain in English ("four score years and ten ago")
- and in French ("quatre-vingts"*
  4*20=80
  , also in the name of the Parisian hospital Quinze-Vingts*
  15*20
  which was originally founded to house 300 patients). In addition, numbers after sixty and eighty go up to sixty-nineteen (soixante-dix-neuf, instead of seventy-nine) and eighty-nineteen (quatre-vingts-dix-neuf, instead of ninety-nine).
Many languages count in little endian up to a certain point and switch to big endian for larger numbers. For example, in English "16" is sixteen but "36" is thirty six.
- Some languages are weirder: Arabic stays little-endian until 100. So "16" is sitta-`ashar (six-ten) and "36" is sitta-wa-thalaathuun (six-and-thirty), but "136" is mi'ah wa sitta wa thalaathuun (a-hundred-and-six-and-thirty). This, however, is a relatively modern construction. Like the language itself, the number were read from left to right; "sitta wa thalaathuun wa mi'ah" (six-and-thirty-and-hundred)
- In older English, it's also not uncommon to see constructions like "six and thirty".
The Dozenal Society of America is the premier American organization for the promotion of base 12.