Daan's answer is correct regarding the two providers named explicitly (+1).
If you want to save yourself some hassle and/or get some guidance along the way, you may want to have a look into one of the dedicated Platform as a service (PaaS) providers targeting the mobile app backend segment specifically. There are probably a dozen more in the meantime, but from the top of my head I recall these three right now (all commercial, but usually some kind of free tier to get you started and/or eternally even for development purposes) - in no particular order:
- StackMob
- Parse
- Appcelerator
The main benefit you'll achieve via this approach is dealing with a platform rather than the infrastructure itself, which abstracts away many (if not all) topics requiring dedicated system administration skills (it usually doesn't hurt to have that knowledge on the team regardless, see DevOps).
The main drawback to look out for is avoiding vendor lock in eventually, i.e. depending on your needs you might want to be able to migrate your app backend to another provider, which is currently not always simple, if at all possible. On the other hand this is likely not your concern right now, if you just want to get started fast and/or develop a prototype for example.
Most of these will use some Infrastructure as a Service (IaaS) provider like AWS or Joyent as their backend in turn btw.
Good luck!
