It’s all a question of numbers: in general terms, between is used when you’re talking about two things, and among is used when you’re talking about three or more thing, whether that’s points, places, people or choices.
So, between means into or across the space separating two points or locations or objects – “He regularly caught the train between Glasgow and Edinburgh”, “He squeezed in between Sally and Jan so he could be in the photograph”, or two things that need to be differentiated or otherwise chosen: “I couldn’t choose between the doughnut and the cream cake, so I had both”.
And among means surrounded by, in the middle of – “I was trapped among the cows in the field” – or indicates a differentiation, choice or division involving three or more parties or objects – “You must choose your holiday destination among Bulgaria, Hawaii and Outer Mongolia”
Between can be used for more than two objects, people, etc. when you are indicating a connection between several entities/parties (“There are links between Spanish, Italian and French”) although this one is a matter of stylistic choice, and I prefer among for more than two, and when a group does something – “We came up with his fine between us: although he got caught, we all raced the shopping trolley down the main road”, although I might be inclined towards among here, too, if I was being particularly picky.
These choices, where it’s not clear which has to be used, is why we editors have style sheets, by the way, so that when we make a choice the first time, we make sure we make the same choice next time.
I’ve been asked about betwixt and between a few times. Betwixt is just the archaic form of between. Betwixt and between means “neither one thing nor the other”
And amongst, like whilst and while, is a mainly British English alternative form of among, but can seem a little over-formal.