Combat steps, attackers and defenders

A combat is three steps (the showdown, the damage, the resolution), and who is the Attacker has nothing to do with whose turn it is. It belongs to whoever made the battlefield Contested.

Worked example. On your turn you move Nidalee, Cat Form (4 Might, "When I win a combat, draw 1") to a battlefield where your opponent has a Hunting Dog (3 Might). Moving there applies Contested, because you do not control the battlefield (rule 190.3.a and rule 190.3.a.1). At the next cleanup a combat is staged, and with the chain empty it begins (rule 462 and rule 463). Step 1, the Combat Showdown: you are the Attacker because your unit applied Contested, you gain Focus as the showdown opens, and Nidalee gains the Attacker designation (rule 466.2.c.1, rule 466.2.c.1.a and rule 466.2.c.3). Step 2, damage: Nidalee and the Dog assign damage to each other and it is dealt at once. The Dog dies and Nidalee survives. Step 3, resolution: a Combat Cleanup heals all units, and since only your units remain you win the combat, so Nidalee's trigger draws a card. You then establish control, and because you had not scored that battlefield this turn it is a Conquer (rule 468.1.a.1, rule 468.3.a, rule 468.5 and rule 468.5.d).

Staging. Combat is staged when units of two opposing players are at a battlefield and its steps have not begun (rule 463). If more than one battlefield has opposing units, the turn player picks which combat goes first (rule 463.1). A combat begins during a cleanup when the chain is empty, a combat is staged, and no showdown or combat is ongoing at another battlefield (rule 462). If a showdown is already ongoing at that battlefield, it becomes a Combat Showdown (rule 462.1).

Step 1, the Combat Showdown. Start of combat effects happen first. Then the Attacker and Defender are established and their units gain the designations (rule 466.2.b and rule 466.2.c). The Attacker gains Focus if the showdown opens with the combat, but if a showdown was already ongoing, whoever had Focus keeps it (rule 466.2.c.1.a and rule 466.2.c.1.b). A unit arriving later gains its controller's designation in the cleanup after it arrives (rule 466.2.c.3.a), and a unit that is no longer at the battlefield loses it (rule 323.2.c). Attack and defend triggers fire when a unit first gains the designation in a combat and go on the Combat Chain, the Attacker's first, then other non-Defender players', then the Defender's (rule 383.4.e, rule 383.4.f and rule 466.2.d.1). Assault and Shield work only while the unit holds the designation (rule 807.1.d and rule 814.1.d).

Step 3, the Resolution Step. The Combat Cleanup recalls attackers if defenders are still present (rule 468.1.a.2). A player wins if they have a designation and are the only one with units there. It is "No Result" if units were recalled, if both players have units, or if neither does (rule 468.3.a and rule 468.3.d). Winning a combat is not the same as conquering. If nothing else is staged there, the player with units remaining establishes control if they did not have it, and this does not have to be the player who applied Contested (rule 468.5 and rule 468.5.e). If no units remain, the battlefield becomes Uncontrolled (rule 468.5.b). Combat then ends and every designation is removed (rule 468.7.a).

Common mistakes. "I'm the turn player, so I attack." Not necessarily: if your opponent's unit is moved into your battlefield, they applied Contested and they attack. "Attackers get Focus." Only when the showdown opens with the combat.