MIDI, which stands for Musical Instrument Digital Interface, is like a standard language. If I'm playing a B-flat on a keyboard, there's a specific code that says that I'm playing a B-Flat, and how loud it is. It's industry standard, which means that I can either play that through whatever onboard sounds the keyboard has, or plug in a chain of electronic modules that'll give me strings, guitars, synths, or anything I want. So I play a note with a MIDI keyboard, which sends it to whatever, which gives me back a B-flat at X volume in whatever I want to do it with.
That's the non-technical way of putting it.
Usually, what people think of when they hear the term MIDI, are songs that sound like they came from an NES. 8-bit type music. That's not always the case. That's really just a default port that comes with most MIDI controllers--you usually add in your own sounds via a VST plug in or maybe a Soundfont. (Believe me, in the 80s, those beeps were hot shit for video games because they didn't take up any space at all. Think of the difference between a .txt file and a .doc file, or even a .pdf file. The information is still there--there are still words, but the other two are a bit prettier, right? More specific with the editing.)
A MIDI Trigger is anything that you can use to create a MIDI signal. Obviously keyboards were the first wide-spread use of it, which eventually lead to the keytar and other chromatic based instruments. And then someone realized that the number of inputs doesn't have to be like a keyboard, so they made drums with it, and eventually...
....you've got something completely new, and we've managed to not only control what note and how loud it is, but any effect that we can think of can be manipulated with those cute knobs and buttons so that -that- note doesn't echo as much as -this- one!
Got it?