"Then" is frequently used to indicate what happened, happens, or will happen next when telling a story. It is very similar to "alors" (at that time) and to "puis". It is also a very good substitute for the ungrammatical "After, ...".
- She started with some small talk, but then she got down to business pretty quickly. The meeting was over in an hour. :)
at that time
- We didn't realize how dangerous it was to cross the river via the train bridge back then. We could have been squashed!
- You should turn off the electricity first; then you can work without worrying about electrocuting yourself.
Compare to "now".
- Now you can find almost any song online somewhere; back then you had to buy a copy of the record to listen to a song over and over again.
now and then: de temps à autre
"when": at what time?
- We didn't realize how dangerous it was when we used to take the train bridge to cross the river.
Entailment is more or less a synonym for consequence.
- Si X est vrai, alors Y est vrai
- If X is true, then Y is true.
if and then are only really used together in formal contexts (math, programming). Most often it's the result of an inference:
- Husband: I'm not feeling well, I'm going to stay home today.
- Wife: Aha! You don't need your car, then? Do you mind if I take it? Mine is making weird noises and I need to bring a bunch of things into the office.
- Husband: I have an appointment with the doctor this afternoon, but I can take yours if you'd like.
- Wife: I'll be back for lunch, and I'll take mine in the afternoon.
- Husband: Ah, ok then, no problem.