Most thing will stack, but there are some exceptions
First, anything that is listed as unique in an item description will not stack. For example, Zhonya's Ring gives +25% AP as a unique. 2 Zhonya's would not increase your AP Bonus to 50%. This also goes for active abilities iirc.
Cooldown Reduction (CDR) stacks but is hard capped at 40%
Dodge stacks, but is not additive (it is multiplicative). For example, if you have 2 itesm giving 10% dodge, they would give (100*.1)= 10% for the first item and (90*.1)=9% for the second for a total of .19% dodge (or you can look at it as each one makes your chance to be hit as 90%, so .9*.9=81% chance to be hit.....or 19% dodge).
Speed has a soft cap. I'm not sure on the numbers, but you get diminishing returns after a certain point. There is no hard cap though, in fact players have broken speeds of over 1200 before iirc (that's about 4x the movement speed of a start character, and about 3x the speed of most people endgame). Boots do not stack, but movement speed from other items (give % movement boost) do.
Aura's are a bit weird as well. You can only be affected by an items aura once from a teammate and once from yourself. So, if I buy a Aegis of the Legion Aura bonus to bothe armor and magic resist), and Larias also buys an Aegis, and we are playing with EncasedShadow, we would get the following
Encased shadow would get the aura bonus only once (he can be affected by only one Aegis aura from a teammate at a time)
Larias and I would both get the bonus twice, once from our own Aegis and once from the others Aegis.
Note that tat doesn't mean you can only be affected by a single aura, just one aura from a specific item. If I were to buy Aegis and Starks, everyone would get the aura bonus from both.
Attack speed stacks but is hard capped at 2.5 I believe
I'm sure I'm forgetting some, but those are the ones off the top of my head.