At a bare minimum, GNU/Linux needs one partition for itself. You can have a single partition containing the entire operating system, applications, and your personal files. Most people feel that a separate swap partition is also a necessity, although it's not strictly true. “Swap” is scratch space for an operating system, which allows the system to use disk storage as “virtual memory”. By putting swap on a separate partition, Linux can make much more efficient use of it. It is possible to force Linux to use a regular file as swap, but it is not recommended.
Most people choose to give GNU/Linux more than the minimum number of partitions, however. There are two reasons you might want to break up the file system into a number of smaller partitions. The first is for safety. If something happens to corrupt the file system, generally only one partition is affected. Thus, you only have to replace (from the backups you've been carefully keeping) a portion of your system. At a bare minimum, you should consider creating what is commonly called a “root partition”. This contains the most essential components of the system. If any other partitions get corrupted, you can still boot into GNU/Linux to fix the system. This can save you the trouble of having to reinstall the system from scratch.
The second reason is generally more important in a business setting, but it really depends on your use of the machine. For example, a mail server getting spammed with e-mail can easily fill a partition. If you made /var/mail a separate partition on the mail server, most of the system will remain working even if you get spammed.
The only real drawback to using more partitions is that it is often difficult to know in advance what your needs will be. If you make a partition too small then you will either have to reinstall the system or you will be constantly moving things around to make room in the undersized partition. On the other hand, if you make the partition too big, you will be wasting space that could be used elsewhere. Disk space is cheap nowadays, but why throw your money away?