One color

In the most common environments, decks with a single color are not competitive in tournaments (due to the fact you’re settling for the top ten cohesive cards in a color instead of two sets of the top five in two colors. The difference in power level between the top 1-5 and the top 6-10 can be massive and game deciding. The other colors also make up for each others weaknesses.) But they do exist, especially in older formats where consistency is required to win in a couple turns and where you have such a large pool of powerful cards to pick from, and are then referred to as “Mono X”.

Mono red
Mono black
Mono blue
Mono green
Mono white

Two colors

Each of these corresponds to a guild of Ravnica. Since the guilds were released (and especially now that we’ve returned) the color pairs have become common names for duo-color decks. Both allied and enemy color pairs are common to see, although allied more so (due to higher deck cohesion).

White + Blue = Azorius
Blue + Black = Dimir
Black + Red = Rakdos
Red + Green = Gruul
Green + White = Selesnya
White + Black = Orzhov
Blue + Red = Izzet
Black + Green = Golgari
Red + White = Boros
Green + Blue = Simic

Three colors

The first of these are the shards of Alara. They jive well because they give two sets of allied colors each. The less commonly used are the wedges, which involve two enemy color combinations: they’re derived from the dragons of Planar Chaos, the volver cycle from Apocalypse, five clans from Khans of Tarkir, or more recently the the triomes of Ikoria. Having only one allied color pair (the two enemies of a single color will be allied) limits deck cohesion, making their use infrequent, although this obviously varies as the meta changes.

Note that if one of the colours is not particularly present in the deck, for example there’s only a single card of a given color, you may see a two color name with a third listed as a “splash.”

Red + Green + Black = Jund
Green + White + Blue = Bant
Blue + Black + Red = Grixis
Red + Green + White = Naya
White + Blue + Black = Esper
Blue + Red + White = Jeskai (clan on Tarkir), Raugrin (Ikoria triome), Numot (dragon from Planar Chaos) or Raka (from Rakavolver)
Red + white + black = Mardu (clan on Tarkir), Savai (Ikoria triome), Oros (dragon from Planar Chaos) or Dega (from Degavolver)
Black + green + blue = Sultai (clan on Tarkir), Zagoth (Ikoria triome), Vorosh (dragon from Planar Chaos) or Ana (from Anavolver)
Green + blue + red = Temur (clan on Tarkir), Ketria (Ikoria triome), Intet (dragon from Planar Chaos) or Ceta (from Cetavolver)
White + black + green = Abzan (clan on Tarkir), Indatha (Ikoria triome), Teneb (dragon from Planar Chaos) Necra (from Necravolver), Junk citation, or Doran citation

Informal usages:

Red + white + black = Borzhov
Red + white + blue = USA/American/Patriot
(although note that Team America is actually black + blue + green)
Red + green + blue = Grizzet
(although it’s usually Simic splashing red)

In addition, it’s especially common for red + blue + green and black + blue + green to be called by their abbreviations — “RUG” and “BUG” — because these are names that are easy to remember and pronounce, even though those are not the proper color orders.
Four colors

Most decks do not have four full colors. As with three color enemies, if they reach this many colors, it’s a shard with a splash of another color. So you’re more likely to see something like “American splash black” instead of “Yore”, and lately simply “four color” is rising in popularity.

Names for four-color identities come from one of two sources:

The names of the Nephilims from Guildpact.
The names of the four-colour “guild identities” defined during Commander 2016’s design.
Reference the one color the four-color combination is missing, thus Non-(color).

The four colour identities’ names are:

Blue + black + red + green = Glint-Eye, or Chaos, or Non-white
Black + red + green + white = Dune (or Dune-Brood), or Aggression, or Non-blue
Red + green + white + blue = Ink-Treader, or Altruism, or Non-black
Green + white + blue + black = Witch (or Witch-Maw), or Growth, or Non-red
White + blue + black + red = Yore (or Yore-Tiller), or Artifice, or Non-green

Occasionally, some informal usages of four color decks, especially when it involves a typical three color deck with a splash of a fourth, will be to include some kind of pun or other indicator to an established name. For example, taking a Jund (Red, Green, and Black) deck and adding Blue would be more likely to make it something like “Wet Jund” than it would be Glint-Eye or Chaos. Or if only splashing Blue for a single card might make it “Moist Jund”, that is Wet Jund that is slightly less wet than Wet Jund. Similar puns can be made for Dark Jeskai (Blue, Red, and White adding Black) and I could envision something along those lines for a splash like Dim Jeskai (Blue, Red, and White splashing Black) although admittedly I’ve never seen it in the wild.
Five colors

Decks with all five colors usually revolve around a single combo that they hope to pull off. It takes a lot of mana fixing and a massive amount of playtesting to get a reliable five color deck. For this reason, you don’t often find them in tournaments. You find them often in Commander (giving the player access to every card ever, greatly increasing the power level of the deck) and in skill challenges where a player just tries to come up with a crazy deck idea to see if he can make it work. Obviously there’s only one five color deck, it uses all five:

Rainbow/Domain/Five-Color/WUBRG (pronounced Whoo-Burg)

Sometimes, four color decks will also be called Rainbow just because they have so many colors.
No colors

Sometimes, particularly in formats with a very large card pool, you’ll see colorless decks as well. The most common name for these is a reference to the old card frame for artifact cards:

Mono brown (not to be confused with BrownTown which is a draft deck leaning heavily on minotaurs)

Some newer colorless decks have come to be referred to as “Diamond” decks in reference to the diamond-like new symbol for colorless mana