In Magic the Gathering 'deck archetypes' are a convenient way to describe how a given deck plays without having to list every card in it. There are 3 broad categories of archetypes, and countless smaller categories that fall within one (or more) of them. Which categories certain decks fall under is a major source of skub. Given Magic's popularity the vast majority of these terms are used in other card games even those that have little mechanically in common with Magic the Gathering. Just about the only TCG term that's widely used across multiple games and is not from Magic the Gathering or older non-collectable card games is calling any combination of cards that let you instantly win the game an "Exodia".
The Big 3
Simple go-for-the-throat rushdown, best exemplified by Red. Red/Black, Red/White, Red/Green and any combination in between are all popular for this. Popular among new and casual players for the simplicity of piloting and the satisfaction of doing big damage fast and dropping lightning bolts on things. Typical ways to describe this strategy include terms such as "turning sideways" or SMORK.
Combo decks rely on one (rarely more) specific combinations of cards that have highly synergistic interactions, ex. Painter's Servant + Grindstone, to derive massive advantage or win the game in one fell stroke. While powerful, they are vulnerable to having their win conditions removed via, for example, Pithing Needle, and typically flounder when denied their combo. Combos exist in every color so there is no particular color archetype to combo decks.
Typically Blue, often Blue/Black or Blue/White, Control seeks to deny the enemy the ability to make moves, whether by countering their spells, destroying their hand, stacking their deck or even eliminating their mana base. Control is often cited as the least fun archetype to play against, and is often the most oppressive in any extended format.
More detailed Archetypes
The Big 3 above are really broad can be further divided into more defined archetypes that cover decks that have the same aim while using different cards. For instance,"This is a Zoo deck." will say a lot more about your deck than "This is an Aggro deck."
Originally, affinity decks used Affinity (a keyword that reduces the casting cost of your spell for each permenant of a certain kind you controlled on the field, be it a specific land or a type of permanent) and similar abilities to reduce the costs of their own cards and drop bombs early. In more recent years Affinity was the nickname for a deck using a plethora of cheap
stupid murder robots artefact creatures like Arcbound Ravager to murder their opponent.
Creature-based aggro to midrange decks using a lot of high value small animals like Kird Ape, Loam Lion, and Wild Nacatl for the more aggressive version, while bigger version would generally include a few creatures that produce mana (mana dorks) to get to 4,5 mana and cast big haymakers such as planeswalkers, dragons or whatever they fancy. If the creatures didn't do the job, a bunch of "burn spells" such a Lightning Bolt or Boros Charm were usually used to finish the job. Almost always RWG.
Runs powerful black creatures with masochistic drawbacks (ex. Flesh Reaver and Phyrexian Negator), in the hope that they can kill the opponent before they kill you.
Examples: Suicide Black
Burn aims to go directly for the enemy's face using direct damage spells and efficient creatures. Burn rarely plays the long game and looks to have an overwhelming advantage in the first few turns before its low-curve cards lose their potency and slower decks come online. Almost always viable whatever the time, being a solid budget option at worst and pretty much oppressive at best (Amonkhet-Ixalan period).
Examples: Red Deck Wins(some variants), Sleigh/Sly
These decks win by placing 10 poison counters on the opposing player (instead of the usual 20 points of normal damage). There are very few ways to remove poison counters and many ways to place them. Poison decks completely bypass lifegain decks and provide a nearly unstoppable win condition. Though poison cards and cards that interact with them have been printed in many sets, poison decks rely almost exclusively on the overtuned Infect and Proliferate abilities from the Scars of Mirrodin block. Despite this limitation they can be incredibly oppressive, coming from behind to place 10+ poison counters in a single turn.
Named after the Millstone artifact, these decks take advantage of an otherwise rarely relevant rule that states you lose the game when you have to draw a card, and you have no more cards in your deck. Often Blue and Artifact focused, these decks use "milling" cards that force the opponent to move cards directly from their deck to their graveyard.
A hyper-focused combo deck designed to do something ahead of curve when an opponent won't be ready for it. Ramp decks tend to rely on either dark rituals, mana elves, or zero cost artifacts and abilities that can use them in some way (improvise from Aether Revolt for example). But as the name implies, these decks are aiming to jump off a ramp and if they don't stick the landing with a good opening hand then they're gonna have problems. Differnentiated from the tron deck below by the absence of the Urza's lands.
These decks use creatures of the same creature type, along with cards that buff that type. Slivers are the most explicitly tribal design in the game. Tribal decks are popular among the more experienced casual players for their high level of flavor, but many are entirely capable of being competitive. "Lord" creatures and Coat of Arms are staples in tribal archetype decks.
Named after Voltron, these decks use the 3 Urza Lands (collectively called Uzratron) to summon powerful creatures (Wurmcoil) or Planeswalkers (Karn and/or Ugin) much faster then would normally be possible. Variants exist using lands like Versuva and Cloudpost, although they achieve the same effect.
Uses a variety of utility creatures to assemble quick combos or toolbox fix creatures. Birthing Pod itself allows for quick creature rotation and tutoring. After Pod was banned, the deck picked up Cord of Calling and Collected Company to fill the gap.