I'll respond under G&K (vanilla was a bit different). For Legalism, you will receive in your first four cities a free (no cost to build, no cost to maintain) culture building that will be the first culture building you don't already have in those cities. Often this means you had a monument in your capital and maybe one other city, but nothing in the last two cities. When you take Legalism, you will get free monuments in any cities that don't already have them and, once you tech Drama & Poetry, you will get a free amphitheater in any city that already had a monument when you took Legalism. You don't even have to have founded all four cities when you take Legalism -- if you found cities 3 and 4 after taking that policy, they will have free monuments when you get around to founding them.
You will hear people talk about "Legalism abuse" -- that refers to delaying taking Legalism until after you have amphitheaters (or even opera houses) in your first four cities, so you get four free opera houses (or museums, although they come so late you have to ask - why bother?). For Siam in particular, the wat is a university replacement that gives culture, so it counts as a culture building for Legalism purposes (I don't think the patch changed that).
Aqueducts with the tradition finisher are different -- you get 4 free (no build cost, no maintenance cost) aqueducts immediately when you take the finisher, regardless of whether you have teched aqueducts yet.
EDIT: What he said.
