Combat Actions.
Potentially you may do about 5 different sorts of thing in combat:
Movement: This may happen before or after your Action, and may be split up to both before and after.
Action: Basically anything your doing in a round. Attack, Cast a spell, Dash (get more movement), Disengage (move away without triggering an AoO), Dodge, Help, Hide, Ready, Search, use an Object. Page 192-193 PHB. Also the ACTIVE use of a skill.
These activities are mutually exclusive, so you only get one per round.
Bonus Action: This type of action is special in that only certain circumstances allow for it. An Off hand weapon, of course. But in that case remember that this is NOT an Attack Action. High level warriors don't get two attacks with their Bonus Action. Certain spells have a casting time of Bonus Acton, allowing them to be done in addition to an Attack or other Action. Spells casting using an Action when you've cast a spell using a Bonus Action is limited to a cantrip with casting time of 1 Action. So, yes, it's possible to cast two spells in a round, if they are the right two.
Reaction: This two is a triggered event, and you only get ONE per round, no matter how many triggers happen. An AoO is a Reaction. The Ready Action used when you're delaying your Action for a specific event sets up a Reaction and it's trigger. Some Spells, such as Shield or Feather Fall are Reaction spells. They take no real time to cast. Yes, this means you might actually manage to get off 3 spells in a round.
Free Action: This isn't really an action category, as in previous editions, so much as something that just happens in combination with Movement and Action. A sidebar with examples can be found on page 190 of the PHB. The limit, at least when I run a game, is ONE such activity. Two would require your Action, preventing anything else (such as an Attack). This will usually only matter when you're trying to pick up two weapons, or a missile weapon AND ammo.