1. Reliability - you want to be sure that your site will be online all day, every day.
2. Versatility - you may want to try out Microsoft FrontPage at some point (it's an excellent aplication, and increasingly popular), and this will mean that you will want your host to install FrontPage extensions on the server. It's easy to do, and the extensions are free, but some hosts make heavy weather of it. Find out in advance.
3. Approachability - can you talk to a human being, or does the host avoid personal contact? Some very cheap hosting companies provide email support only, and it can be very frustrating to wait days for the answer to a technical question.
Make sure that there are no unduly restrictive bandwidth limits - what's the point of launching the new 'Friends reunited' and attracting a million visitors a week if your web host limits your file transfer to 500Mb a month? Additional bandwidth charges can be crippling, so find out before you commit.
Don't worry too much about the differences between the various server types - they'll all run your site quite happily - but do make sure that the host makes adequate security arrangements. You don't want to be hosted on a machine that runs unattended in an unlocked boxroom for weeks on end.
Most hosting packages will provide far more server space than you're likely to need, and this isn't usually a worry. If you register a domain with a host, make sure you get written confirmation that you are the domain registrant. Some unscrupulous hosts have been known to register customers' domains to themselves, and this makes life very difficult if you decide to move your domain to a new host.