Most accounts have time limits associated with them. You may be allowed 3 hours per day, or perhaps 50 hours per month, or whatever, and are billed extra if you are logged in more than that. Accounts without time limits are useful for people like me who spend hundreds of hours on-line each month, but they are generally more expensive.
Some low-end accounts use a menu system (like many BBS's) which allows you to only use (for instance) mail and news, and maybe some sort of on-line chatting, like IRC (worldwide CB-style) or talk (normal Unix individual chat sessions). Simple, and probably good enough if you aren't a Unix geek and don't want to be.
Medium-grade accounts are standard login Unix shell accounts. With these, you get a login shell just like you get when you login to your Linux machine (more or less). You generally have full access to all the programs on the machine, and usually have a few megabytes of space (5 seems to be popular right now) in which to store personal files. In addition, many of these accounts allow you to temporarily exceed the space limits. You are usually allowed to compile and run programs of your own, and very often standard Unix programs such as TeX are provided. At least at this time, most Linux users will probably want this type of account.
The best (and usually most expensive) personal accounts allow your machine to be an actual node on the Internet, using either SLIP or PPP, two protocols for turning a serial connection (such as your modem allows) into a TCP/IP connection. The good side of this is that your machine is directly on the Internet, and once you are connected you can run many simultaneous connections over one modem. The bad side is that your machine is directly on the Internet, and without proper security measures, crackers can gain access to your computer and your files, either stealing data (e-mail or copyrighted programs on your mounted DOS partition, for instance) or wreaking havoc on your system. Choose this option with care, and consider getting a book on Unix system security, such as Practical UNIX Security, by Simson Garfinkel and Gene Spafford, published by O'Reilly, ISBN: 0-937175-72-2.