What kind of Nikon Lens should i use?

I was looking into getting some sport photography and day-day use Nikon lens for my D90. Please could you tell me what kind of lens can produce this quality photos.

https://toughmudder.com/content/2014-capital-region-sunday?page=1
https://toughmudder.com/content/2014-wisconsin-saturday?page=1
https://toughmudder.com/content/2014-capital-region-sunday?page=9
https://toughmudder.com/content/2014-wisconsin-saturday?page=8

Questions like yours annoy me, don't rush off in a huff though.
The shots in your links are made by a variety of lenses, some are wide angle and other are more at the telephoto end. You really need to decide on what you want to achieve, the type of result you want and then take it from there. Your best bet would be to give the local library a visit and read through a couple of photography guide books, photographers know which lens will give them the effect or result that they want, that's just as much a part of photography as pressing the shutter release.
I've covered fun runs with a 90mm lens and the results were decent, you should really be working out what your own preference is, not asking a bunch of strangers what theirs is.
I'm currently in the market for a 70-200 F2.8 for covering fun runs, here's a typical shot that I've taken with basic kit, it's not going to win any prizes, it's just to let you see that I actually do know what i'm talking about.

You could use the following focal lengths to get photos like those you linked to:
24-105
28-105
18-135
18-200

Sport photography is not day-to-day. You would need a long telephoto for that, at least 200mm. The most popular is the 70-300mm. If you can afford longer and/or faster, the better.

For day-to-day use, stick to the 18-55mm. It's more useful for general shooting.