Can you try answering "using lots of terminology"? It might be helpful, even for someone who doesn't understand all the terminology.
OK -- how about just a little for starters.
First of all, the basic thing is something that doesn't need any terminology:
You need to have some basic feel for music (not necessarily
classical music!) and to have at least a little bit of a decent ear (which I'm sure everybody or almost everybody on this site has), and to use some judgment. Because, even if someone knows all the "terminology," they still need to have a sense of where the harmony needs to go in order for there to be a feeling of finality or "arrival."
But, without some knowledge of music theory, particularly about common kinds of chords, it would probably be a little hard for most people to just use "ear" and judgment to know what key is going on. They
could do it, but it would take trial and error -- i.e. trying every which possible chord to see which one gives the feeling of "arrival." Some people are probably exceptions; like, people who are very good at playing by ear might be able to just "know" what chord that is, without knowing what the chords are called or anything else about music theory.
As most people here probably know, chords are labeled by Roman numerals, from I through VII. Each kind of chord has a certain "feeling" to it, and most of us have
some feel for the main ones, regardless of knowledge about music theory: I, IV, and V, which are also called tonic, subdominant, and dominant. There's an extra kind of chord that's pretty common, which doesn't really have a Roman numeral (although we often make one up for it, and I imagine some people reading this might want to say "sure it has a Roman numeral" and I'd feel like arguing back but I won't)
-- "diminished 7th." Anyway.... Those are the most common chords, and
they happen to be important in most modulations. If you have a feel for those chords -- i.e. how they sound and how they 'feel' -- then you have a big head start in figuring out the key of any passage. It's kind of like this: IV "wants to" go to V, which in turn "wants to" go to I -- and whatever feel like "I" in any given passage,
that's the key. (Diminished 7th's can go just about anywhere, so they don't tell anything about what key you're in, at least until you see what the next chord is.)
One final thing: It's not always that simple. Sometimes it's ambiguous whether it's a "key" at all, or just a passing thing that doesn't change the basic key -- and very often, different people would have different opinions of it. To a large extent, it's a thing of the person's general taste about how prominent a new harmony needs to be in order to consider it a "key"; like some people might generally need it to be at least a few measures' worth, other people might feel there to be a new key even if it's just a moment, depending on how strong it is.
But, even if the people might argue about whether it's a new "key" or not, usually there wouldn't be any disagreement about a new chord or harmony having become somewhat of a center at least for a moment, and they'd agree on what chord that is.
BTW, here's an interesting sidelight: Sometimes the main chord of the new key (i.e. I, "tonic") never gets played (!) but is implied strongly enough that it's a new key anyway.
I hope that's helpful.
