In practice, you're correct and it means that it's a fixed focal length lens.
Historically, the term "prime" derives from cine work, where it had a more literal meaning of the primary lens of a multi-lens system.
The concrete advantages are typically that they are much more compact, and have a much higher quality/cost ratio, as they're easier to engineer to a high degree of correction. There is no such thing as an f/1.4 zoom lens, for instance, but f/1.4 primes exist in several focal lengths.
Zoom lenses are catching up on the quality front, a particular example being the Nikkor 14-24, which is reputedly as high-quality as the primes of similar focal lengths (and the Nikkor 24mm is considered a classic design).
The more subjective pros/cons are – obviously – a lot more subtle, and vary from person to person. For example, some people find that primes let them focus more immediately on the composition of the image and/or their surroundings (i.e., it's one less camera setting to think about), while others prefer the ability of a zoom to quickly change the field of view, producing extremely different compositions in quick succession. The same person might like both in different situations. I think the only fair answer to this aspect of the different approaches is to try using a prime regularly and see how you get on with it.