No, it's similar but not the same as compression. The thing they have in common is that there's a comparison between the signal and a reference.
In compression, that comparison is continuous; you might think if it as a "mode". As long as the signal is high, you get compression. Further, it's quite important how far above the threshold the signal is - that's kinda the whole point.
In a trigger, none of that applies, other than than that there is a comparison. Other responders here have used the word "event." It's a single point in time, meeting the specified conditions. The "normal" conditions are (traditionally) a specific signal level, and the slope of the signal when that level occurs. Digital scopes can provide a much wider variety of conditions just by having the processor take a look at the record of the signal. (Since it's in memory, we've got plenty of time to analyze it.)
In any case, the result of recognizing the event is to scream "NOW!" (or in digital land, "HERE!"). A single point in real or virtual time.
Other elements of the scope are then responsible for displaying the signal starting at or around that single point. That stuff has no idea how the trigger event was generated. It starts. It also says "Busy drawing now, no new triggers for a while, please!" (This is "holdoff.") Eventually it says "Okay, start looking for another trigger event."
Triggers are useful in devices other than oscilloscopes. For example, I have a signal generator that can use triggers for several purposes, all based on "NOW!" - start a single wave, start a burst of them, synchronize the wave's starting point, etc.