You get to pick whichever server you want to play on when you get access to the game. (you can also make characters on multiple servers)

Currently There is nothing that will show your character on the web site (I think).

Make sure when you choose to pick the proper timezone (ie: you live in the west pick a US west server, otherwise noone will be on when you want to play)

Good info Corada but what I'm assuming OP means is they have a char on a server already but might not remember name, and now wants to tell friends the server name.

I ran into a similar prob in game, I could not find anywhere the name of the server without logging out to the server selection screen.

If you were a members of a guild that was part of the guild deployment process, and had at least 4 members (who had pre-ordered) in the guild before Dec. 2nd when that phase of guild deployment closed, then you should have received an email a few days ago sending you to a website that would tell you which server your guild was deployed to.

That website is right here.

It is also true that if you were part of this process, then when you finally get to log in and get to the server selection screen for the first time, there should be a yellow flag next to the server name that your guild was deployed to.

Nothing says you HAVE to play on that server, you are always free to play on any server you want, by the way.

If you were NOT a registered member of a guild (that was part of the SWTOR.com guild deployment process), or if your guild did not have a minimum of 4 members who had registered pre-order codes before Dec. 2... then your guild was not deployed. That means you need to talk to your guildmates some other way and figure out what server you want to play on, and then create your guild in-game (which you can do once you reach your faction's fleet station, right around lvl 10).

If you are not part of a guild at all, then just go ahead and pick any server you want based on whether it is PvE, PvP, time zone, etc.