What are the most common Protoss Fast Expand builds and what are the advantages and disadvantages of each?
I could have sworn I answer this question once before... so let's focus just on the Pros and Cons part.
There are two major FE builds seen in top level play:
There are a couple of players who will favor a 2 Gate Expand, but for the purposes of illustrating the Pros and Cons I'm going to ignore that for the second. Suffice to say it gains some measure of the Pros and some measure of the Cons.
1 Gate vs 3 Gate
It is important, when discussing the number of Gateways to build, to establish that we are not talking just about building this many Gateways, but also keeping up production out of them. Thus a 3 Gate build is not just an extra 300 minerals over a 1 Gate, but also the cost of building units out of those 2 extra Gateways. If you are going to build extra Gateways and then not use them, then they aren't serving a real purpose are wasted money. Like wise a building an early Nexus and then not building workers for it is just as big of a waste, thus a 1 Gate isn't just 400 minerals behind a 3 Gate in Gateways.
The trade off in a 3 Gate vs a 1 Gate is a later expansion for stronger early pressure. By choosing to go 1 Gate a player has few defenses against an aggressive opponent. Even a Macro Zerg might do an early Roach push if he senses a weak 1 Gate opening. By comparison a 3 Gate is a much stronger force with which you can pressure your opponent. Again, if we consider the Macro Zerg or 3 Rax Terran, opting for a 3 Gate allows you to pressure your opponents expansion or successfully defend against early pushes. This is a much "safer" build, but you pay for it in the long run.
The reason a 3 Gate is so very safe is that it is hard to tell the difference between a 3 Gate and a 4 Gate or 3 Gate Robo. As a result the 3 Gating Protoss has the advantage of looking like they're going to make an early push. This will force many opponents to over produce units at the expanse of Tech (Terran) or Workers (Zerg). It can even be used to break a 1 Gate Protoss.
By comparison the 1 Gate Protoss relies on dying early scouts and getting up an early forge to supplement his defenses. While a Photon Cannons aren't held in high regard by many players this reduces the dependence on gas and allows for heavier mineral saturation that serves double purpose in getting the Expansion up and running.
As I mentioned before a 2 Gate is somewhere in between, however, it does not have the benefit of being mistaken for a 4 Gate or the lower dependence on Gas. Some people still believe it provides enough of an early defense to deal with early Harassment (think Hellion/Reaper/Speedling) without delaying the early expansion unnecessarily. This decision is one that comes with practice.
But Which One?
Usually the decision between a 1 Gate and a 3 Gate is a sign of your comfort level in dealing with early pressure. If you feel very capable in disaster management, have good micro, and are against a more passive opponent, then 1 Gate is definitely the better decision. However, if you want a safer build with the option to switch to a one base all-in, a 3 Gate is a very powerful choice.
Ultimately, the decision is one of comfort and practice.
See here for a lot of protoss' build orders.
And in particular:
Be aware that every FE build order is very weak against cheese and early aggression, so be careful and scout.
Forge expand vs zerg as seen on day9 http://blip.tv/file/4585419
Note: Mostly chronoboost probes