The chain, priority, and focus

Two checks decide whether you can play something right now. First, what state is the turn in? Second, do you hold Priority (and, in a showdown, Focus)? Everything below is those two checks.

Worked example. On their own turn your opponent plays Blast of Power, an Action spell that kills a unit at a battlefield. The moment it is played a chain exists, so the turn is in a Closed State and only cards and abilities with Reaction can be played (rule 309.1.a). You hold Brawl, an Action spell, and Abandon, a Reaction spell that counters a spell. Brawl is illegal: Action adds showdowns, not Closed States (rule 806.1.c.1). Abandon is legal, because Reaction does everything Action does and also works in Closed States (rule 813.1.b and rule 813.1.c.1). Abandon joins the chain Blast of Power started, since only one chain exists at a time (rule 330.1 and rule 330.2), and the newest item resolves first (rule 340.1). It would resolve before Blast of Power once everyone passes.

The four states. The turn is either Neutral or in a Showdown State, and either Open or Closed (rule 308.1, rule 308.2, rule 309.1 and rule 309.2). The combinations are Neutral Open, Neutral Closed, Showdown Open and Showdown Closed (rule 310). By default you may play cards and activate abilities only with Priority on your own turn in a Neutral Open State (rule 310.1.a). A Showdown State allows only Action or Reaction (rule 308.1.a), and a Closed State only Reaction (rule 309.1.a). Ambush is shorthand for a unit that may be played to a battlefield where you control units and has Reaction while it is (rule 822.1.b). Priority is the exclusive right to take Discretionary Actions, the actions you may take at will on your turn in a Neutral Open State (rule 312.1 and rule 411.1.a), so a Standard Move is off the table while a chain exists (rule 144.1.b).

How a chain runs. An item you play is first Pending, then Finalized, and finalizing does not pass Priority (rule 329 and rule 337.1.a). A unit, a gear or an ability that Adds resources resolves immediately after it is finalized, so it cannot be responded to (rule 337.2), but a Play Effect it causes still goes on the chain and can be (rule 383.4.a.2). When nothing is left to finalize, the controller of the next item gains Priority (rule 337.4). They may play a legally timed card or pass (rule 338.1). If every player passes in sequence without adding anything, the newest item resolves (rule 339.1), and if items remain, the controller of the newest one gets Priority again (rule 340.4).

Showdowns and Focus. A showdown is a window in which players alternate playing spells (rule 342). Focus is the permission to act in a Showdown Open State, and you need both Focus and Priority to do so (rule 313.1 and rule 313.4). Nobody has Focus in a Neutral State (rule 313.5). The player who made the battlefield Contested gets Focus first, and gaining Focus gives Priority (rule 345 and rule 313.2). When the last item of a chain resolves during a showdown, Focus passes to the next player (rule 346), unless a triggered ability or an Add ability opened that chain (rule 346.1). With no chain you may pass Focus, and if every player passes in sequence the showdown closes (rule 347.2.a and rule 348). In a non-combat showdown, a player who is alone at the battlefield then establishes control (rule 348.2.a).

Common mistakes. "Action means I can respond" is wrong: Action reaches showdowns, and only Reaction reaches Closed States. "I passed, so I lost Focus" mixes up two passes: passing Priority while a chain exists keeps your Focus (rule 313.3), while passing Focus is what you do in a showdown with no chain (rule 347.2).