I have heard these words used a lot in Matrix documentation. Here's what I think they mean:

• Bot: Like a Discord or Slack bot, can be run off of a normal account on a homeserver (no homeserver hosting required)
• Bridge: Connects Matrix and another messaging service, self-hosted homeserver seems to be required
• Integration: Another name for a Bot, or is it something displayed in Matrix clients (ex: Element has n Etherpad integration)???
• Appservice: A better bot???

There's also the whole language around bridges (puppeting, plumbing, etc.), but that's for another question.