What are the advantages of a prime lens and lower f stop number?

I recently purchased a Nikon D3100 with the kit lens and a 55-200mm lens.
However, a friend recommended that I invest in a prime lens with a higher aperture value.

If I were to buy one I think I would be purchasing the AF-S 35mm 1.8

However, I was wondering what the real advantage of this lens would be?
What are the advantages of any prime lens? What are its practical uses?
Also, what are the real advantages of a lower f stop number?

I know a lower f stop number will create a more shallow depth of field, but would this make a drastic difference in the final image in comparison to the 3.6 available on my kit lens?

If someone could clarify on this topic, I would greatly appreciate it.
Also, if there are any other differences, feel free to make mention of them. I'm really getting into photography recently and wouldn't mind learning something new. :]

Your kit lens is only f3.5 at 18mm, it goes rapidly downhill from there. At 35mm it's about f4 I think. That makes a real difference in the amount of light you can let in and in how shallow your DoF can be. Play around with www.dofmaster.com to get a feel for that.

Any prime will be very sharp and have good contrast, that's because there's far few design compromises in it. It also forces you to think more about your compositions because you can't use zoom to save the day.

Personally I'd advice you to get the hang of the camera first. Additionally, Nikon will soon release a AF-S 50mm f1.8 as well. Now, which one is best depends a lot on you. What I always say to people wanting to buy their first prime is to take a piece of masking tape or another tape that comes off easily without leaving muck and to "lock" the zoom ring on either 35 or 50mm. That way you will get a feel for which focal length will be best for you.

Why 35mm? What would you use this lens for?

If for everyday, walking around, then 35mm might be perfect for a prime. If you're interested in landscapes though, you'll want something much wider, like between 10 and 20mm. For portraits, I wouldn't get a prime any shorter than 50mm as you want a good working distance from your subject, and up to 100mm.

There's the 50mm f/1.8 at about $125 (the current version won't auto focus on your camera, but an AF-s version would).

Now, why would you bother with an f/1.8 when the 55-200 is f/3.5 at 55mm? Well it's because very few lenses perform well wide open. I very rarely shoot with my f/1.4 wide open if I can help it, but often at f/2.8 to f/5.6 because it gets sharper as you stop it down. If you start at f/3.5 and want to stop down about three full stops, you're approaching f/8 which generally isn't going to allow for shallow depth of field. That's fine if you're shooting landscapes but you may not want that much depth in a portrait. The better the glass, the less you need to stop it down to get really sharp results too.

Bottom line, if you're interested in landscapes, consider getting the 18-55mm to start with. And if you really want a decent portrait lens, consider the 50/1.8

A prime lens has less potential distortion and light loss.
A lower f stop lens allows taking non-flash pictures at lower light levels at greater shutter speed - I used to enjoy grabbing back stage theater pictures at 1/60 or 1/30 with an f/1.2 lens

As well as allowing a greater depth of field than your kit lens, the wider aperture will allow more light in, which will in turn allow for shooting on lower light conditions. There's a big difference between f1.8 and f3.5 - that's over 2 stops more light.

Prime lenses tend to be better optically as its easier to construct a non-moving lens system.

I would recommend the 50mm f1.8 as your first prime - ideal general & portrait lens. You just have to remember to 'zoom' with your feet!

Prime lenses are lighter, cheaper, and can have a larger aperture than a zoom. A larger aperture means it allows more light in. 1.8 is a full two stops faster than your kit lens at 3.6.

There are some people who will say that a prime lens is sharper than a zoom, but they aren't comparing apples to apples. The prime you are looking at will definitely be a lot sharper than your kit zoom, but not as sharp as a Nikkor f/2.8 ED zoom.