I've shot Nikon for about 6 years. I'm writing this assuming that you have a D3xxx or D5xxx, but not much would change for any other crop sensor camera.
I'm going to start with the tl;dr: You can get good third-party lenses, but you're almost always going to be compromising on something, unless you're paying big bucks for them in which case you may have well gove Nikkor.
Now for the long answer:
What are you going to be shooting? Many new photographers think that they need a lens big enough to see their subject's nose hairs through the viewfinder. If you're shooting running/track you really don't need more than 200mm, maybe 300mm. If you're set to capture large and fine photos, you can crop and blow up anything in the frame. Even the 18-55mm lens that comes with the camera can get you pretty close. With lenses this long, make sure they have VR, or the Tamron/Sigma equivalent. A monopod/tripod is very useful and is a requirement at longer focal lengths.
Aperture is very important for shooting sports, and this is where you'd find the biggest improvement over your kit lens. If you're shooting sports, or anything else in motion, and you want to "freeze the action," you need a fast shutter speed. For running, 1/250s to 1/500s will probably be good enough, other things will need faster speeds. In order to let enough light into the camera, you need a wide aperture (smaller f-number equals wider aperture). For sports I'd look at f/4 at least. If you're shooting inside, you'll probably need wider. On a nice sunny day you'll probably be shooting f/8, so any lens will get you there. If you don't have a wide enough lens (often called a fast enough lens), you'll have to increase the ISO, which is fine to a point, but will increase the noise (speckles) on the photos.
I used this for a long time for all sorts of photography, including indoor and outdoor track.
http://www.adorama.com/NK55200VRU.html
It would be fine for most situations.
If you struggle with having enough light, this would be better
http://www.adorama.com/NK702004U.html
If you wanted to get real crazy, then this would be the way to go
http://www.adorama.com/NK70200AF2U.html
The only real difference between those lenses is the aperture. The f/2.8 will allow for fast shutter speeds in pretty low light.
With third-party lenses, you're often giving something up because the lens manufacturers don't know the proprietary information about Nikon cameras, and Nikon camera software won't necessarily know the lens specs. This can lead to distortion or in rare cases complete incompatibility.
If you have any specific questions, let me know and I'll try to help. I'm sure there are way more experienced photographers than me around here that will chime in too.