Use Windows Authentication unless you have a reason to use mixed (SQL Server + Windows). Admins on the local computer will be members of the sysadmin role - can perform any operation (e.g. root access).
There is also a sa SQL Server logon that is a member of the sysadmin role - this is created regardless of the authentication type, but you can only use it if mixed mode authentication is enabled.
Windows - You are authenticated automatically based on your Windows/Domain user account. SQL Server - You have to supply a username and password.
Sorry, forgot to mention - it's possible to change the authentication mode after you have installed SQL Server. If you use mixed, be sure to use a strong password for the sa account - if you are developing applications, don't use the sa account in you application. Create another account with just enough permissions to perform its functions.
I've not tried the advanced services version, but the management studio express is almost certain to work with it. Also, I suspect the install might include managment studio express rather than a seperate install - just a speculation...