On a fresh Amazon EC2 "small" instance, as of 2010-05-31, the SQL services are disabled by default.
Fix: Enable any service starting with "SQL".
- "Start..Administrator Tools..Services".
- Select every service starting with "SQL", set the startup type to "Automatic".
- Start every service starting with "SQL".
- Some services will refuse to start, don't worry about this.
Now, you can go back into "SQL Server Management Studio", and everything will work nicely.
Update
2010-05-31. I tested the instructions above, with a fresh Amazon EC2 instance with SQL Server installed, and everything worked just as I described.
Update
2010-10-07. The real reason Amazon EC2 disables SQL Server, by default, on a small instance is that it doesn't actually have enough RAM to run things properly. I'm not even sure why they installed it in the first place. From bitter experience, I'd recommend that you run MySQL, given the choice, as its faster and lighter, or upgrade to a decent sized instance.
Update
Another reason why SQL is disabled, by default:
I'm pretty sure they they want to avoid being "that" company that lets loose another
"code red" or "sql slammer" worm - @Dan Esparza