There are four basic ways to make your opponent lose the game:
Typically, most decks are designed to reduce any and all opponents to 0 life. Decking and alternate win conditions are rarely used in constructed tournaments, though there are notable exceptions like Battle of Wits and the Tolarian Academy + Stroke of Genius combo deck made popular by Urza's Saga. On the other hand, in Limited play using 40-card decks, decking can be a viable win condition, depending on the card pool. In casual play, anything goes.
Once the player has chosen a victory condition, he or she must choose how to achieve that condition via a strategy. Most strategies can be described by three different terms: Aggro, Combo, and Control.
The Aggro, Control, and Combo strategies can be further combined to describe archetypes, which are blueprints for decks based on Magic theories that have been successful in the past. However, for a game as complex as Magic, there are always exceptions to be made and rules to be broken. Many decks contain elements of all three strategies, so one must keep an open mind on what constitutes an effective deck.
Aggro ("aggressive") decks attack the opponent by producing maximum damage output in the shortest number of turns, usually a fast and numerous swarm of creatures. They seek to win quickly or not win at all. Aggro decks typically have little interest in anything except reducing the opponent's life total to zero.
Red is the most aggressive color, with white and green's cheap, efficient creatures also contributing well to the Aggro archetype. The "suicide black" deck style also fits into this archetype.
Some players argue that green "Token" decks are superior to red aggro decks.
Control decks seek to enforce the pace and rules of the game. They are reactionary and often extremely disruptive to the opponent, protecting its resources and prolonging the game at all costs. Control decks require a large amount of mana over many turns to build up control over the game, and eventually become unbeatable via an academic win condition (a creature or possibly a combo). Most control decks exercise the concept of "card advantage;" that is, drawing more cards than the opponent, or making uneven card-for-card trades. Blue and White together are the classic Control colors, though Black is also more than capable.
Combo decks typically seek to accomplish one of two things - assemble a small set of cards that generate a powerful combination; or generate a large amount of mana to play one massive game-winning spell (this mana is often generated by a small set of cards). Combo decks can vary wildly; they can be fast or slow, fragile or resilient, and relatively simple or extremely complex. Combo decks are famous for killing the opponent in a single explosive turn after some preparation time.
Combo decks run the gamut of colors and power levels, though Blue and Black traditionally provide excellent tutors (ways of searching through one's library) for finding combo pieces quickly.
The Midgame archetype is characterized by a high mana curve with relatively large creatures. Midgame typically employs mana acceleration along with some creature-control elements to aid its gameplan - to reach the midgame where its powerful threats can dominate.
Green is an excellent Midgame color, providing mana acceleration and big, threatening creatures. White, Black, and Red also have large threats (such as Angels, Demons, and Dragons) and mass removal at their disposal.
Aggro-Control (often referred to as "Tempo") is a hybrid archetype that contains both aggressive creatures and control elements. Aggro-Control's gameplan is to quickly play threats, then protect those threats by disrupting the opponent long enough to win the game.
Blue's countermagic and card drawing are typically the essential Control parts of Aggro-Control. The Aggro part comes from efficient creatures in any color.
Combo-Control is an incorrect, ambiguous term that could mean a Control deck that finishes the opponent with a combo ("Control-Combo", which is simply regarded as a Control deck) or a deck that proactively uses Combo pieces to control the game ("Combo-Control"). Combo-Control decks are better known as Lock decks or Resource Denial decks - decks that prevent the opponent from doing anything useful. These decks are typically named after their combo or the resource they deny. Library destruction, land destruction, and hand destruction are typical, but the most famous is "Scepter-chant," which forbids the opponent from doing anything whatsoever.
Aggro-Combo is a relatively rare archetype that usually stems from a typical Aggro that has a notable, but non-critical combo. For the sake of simplicity, Aggro-Combo decks are usually regarded simply as Aggro decks with a "trick" that can suddenly win the game. They can be dangerous due to the fact that this can give them an edge against simple Aggro decks while giving them some possibility of matching speed against Combo decks.
Ultimately, some Magic decks can even claim to do basically anything. By utilizing insane engine cards, a large "toolbox" of silver bullet cards, or playing only the best cards for sheer power level alone, decks that can claim to be all archetypes at once are usually both adaptive and unpredictable in nature. Lacking the speed of an aggro deck, the disruption of a control deck, and the focus of a combo deck, the extremely rare Aggro-Control-Combo archetype attempts to make up its shortcomings with adaptability.