Archetype

A Magic Archetype is a recurring deck or strategy with many possible variations. Archetypes are defined when they have been prevalent in several tournaments and have showed results repeatedly, Top 8 or higher. They should also be an idea that is playable in many formats, rather than just a pile of cards that wins. Traditionally, the three essential types of Magic decks are control, aggro and combo. Each of these has many variants.

Rock, Paper, Scissors of Magic
The main archetypes Aggro, Combo, and Control form the rock, paper, and scissors of Magic: The Gathering. Aggro tends to beat control because it develops an advantage before control can play its signature cards. Control tends to beat combo because it can disrupt the most important pieces of the card combo, leaving the combo player with weak cards. Combo tends to beat aggro because the combo player can finish their combo, killing the aggro player, while the aggro player is still fighting towards victory. Because of this tendency, elements of aggro, combo, and control are used by wise players in order to build the most effective possible deck.

Hybrid Archetypes
A hybrid archetype combines two archetypes to help reach the end game. The most commonly used hybrid archetype is Aggro-Control, combining the controlling fact of control and then Aggro for aggressive. You control the battlefield and attack aggressively to end the game. Both of the two archetypes help you reach the end game and hopefully, your opponent can't answer the threat that you bring.

Defining Archetypes
New (sub-)archetypes are first defined when a rogue deck enters a tournament setting such as a PTQ. The deck places in the top 8, often top 4, and the list is put on the internet. People, seeing the high finish and the wacky deck design, will copy it. This is called netdecking. The deck establishes a name for itself, usually something funky, but it is not an archetype yet.

Next, people in other formats notice that the deck's unusual name and power level have really stirred things up. They decide to try the deck out in the format that they play, leading to multiple decks spread over different formats. The funny thing is, all these decks have the same name, roughly the same strategy, and often some of the same cards. Now, it has truly been transformed from a rogue deck to an archetype.

Control
Duck and Cover plays one very potent threat (the Duck). Usually it is a very large creature often with an evasion ability, such as Leviathan or Dream Trawler that it can use to bash face to end the game. The rest of the deck is the cover in which the deck abuses counterspells and card draw to make sure that opponent's threats never stick and the Duck stays in the game.
 * Duck and Cover Fatties with backup

Land destruction hits the opponent at its weak point mana. Almost no deck can function without a stable mana base, and by destroying lands and artifact mana, this kind of deck wrecks the opponent's ability to even play spells. Answers require mana, and mana usually requires lands. Solution: Hit those lands. Usually green, white, or black based, but red works as well.
 * Land Destruction Mana Denial

Discard makes sure its opponent never gets to play out their plan, first by emptying the other player's hand and then by making sure it stays that way. Discard control decks are usually base black, often with blue or green for backup. It kills with a combination of enchantments like Megrim, artifacts like The Rack, and creatures like Hypnotic Specter. Once this deck has control of the game, there is almost no comeback.
 * Discard They can't play spells they don't have

Aggro
Full of lots of large, undercosted creatures and burn spells, sligh looks to win by sheer overwhelming beatdown at an unbearable pace. This falls under the category of tempo based decks, but only to an extent. This deck is hellbent on killing as fast as possible without using combos or expensive spells, and is seen in many different formats under many different names.
 * Sligh Hit 'em hard and fast

Based almost purely on burn, this deck looks to win fast and big, by keeping threats on the board and backing them up with direct damage. Easily confused with typical sligh, RDW keeps spells like Lightning Bolt at the front line, so its powerful dudes can connect uncontested. It is named Red Deck Wins because it is fast and furious; and most notably, red.
 * Red-Deck Wins Burn and beat to victory

Combo
This archetype is for a deck that wins by playing a lot of spells, to increase Storm count, and then finish with a powerful Storm effect such as Tendrils of Agony, Grapeshot, Empty the Warrens</c>, or Brain Freeze</c>. These decks can contain infinite combos, but are generally only worried about reaching a set goal where the spell will be lethal.
 * Storm Play lots of spells, then finish 'em

This name of this archetype is a misnomer, as there is rarely an "infinite" loop in Magic, such loops end in a tie. Instead, these decks are often made of a few combo pieces, that will guarantee either a win or the time to do so. In addition, these decks often include tutors, card draw, and enough removal to ensure that it will stay in the game long enough to combo off, and win. These decks tend to be slower, but when control elements are added, they can do very well. Examples are the Vintage Bomberman deck, and the Standard Reveillark Combo deck.
 * Infinite Combo Play all the combo pieces, and win

Hybrid
Also known as Vintage Fish deck, this deck uses control as a backup to prevent its opponent from progressing while hitting with little creatures, White weenie style. Main strategies include mana denial, Standstill</c> and Mystic Remora</c> type effects, removal, and bounce.
 * Aggro-Control Weenies with backup

As creatures have gotten more efficient and planeswalkers exist to punish board wipe effects, midrange has evolved as a pivot between aggro and control and can play either side of the matchup as its draws see fit. The ratio of creatures to spells vary, but typically the creatures are of the highest possible quality and generate significant value upon entry, allowing it to add to the board and win against slow starts but still maintain card parity against removal.
 * Midrange

Draft archetype
Draft archetypes are archetypes, which are designed to emerge during draft. In contrast to the above archetypes, almost all draft decks are a collection of creatures and removal spells, resulting in varying flavors of midrange.

Usually each pair of colors has a distinct archetype assigned to it, which is signaled by an uncommon gold card. Thematic archetypes are usually shown through certain approaches to mechanics, or usage of tribal. Generalized strategies that exist are go-wide tokens (W/x), fliers (U/x), sacrifice (B/R), ramp (G/x), spells-matter (U/R), control (U/x), recursion (B/x), and neo-ferocious (R/G).