When players can perform actions, cast spells and play abilities is determined by a system of priority. Only one player can have priority at a time. The active player (player whose turn it is) will receive priority at the beginning most steps and phases of his or her turn. The steps where this player does not automatically receive priority are the very first, the untap step, and the very last, the cleanup step.
When the active player receives priority he can do one of three things; perform a special action (like playing a land in a main phase or paying a morph cost to turn a face down creature face up) play something or pass. If he performs a special action or plays a spell or ability he or she will receive priority again.
If a player casts a spell or plays an ability it is assumed that unless he or she says anything that he or she is passing priority immediately afterwards. Basically it is up to this player, he can pass after playing this spell or ability or he can keep priority. The important thing is to be clear.
The third option that the active player has is to pass priority. Basically this means that this player does not want to do anything.
Once priority is passed, then the non-active player can play spells and abilities or perform special actions. It work the same for the non-active player just as it does for the active player.
When both players have passed priority with nothing on the stack, then the game will advance to the next step or phase.