Will this lense fit with a Nikon d5000?

Will this lense

http://www.digitalrev.com/product/sigma-18-50mm-f-2/MTAwMjE0Ng_A_A?googlebase=1&country=GB

fit on the Nikon d5000.

I'm looking for a lens better than the stock lens that comes with it?

If it does fit, in what ways is it better than the stock lens?

I'm very new to this, please make answers simple. I can't understand all this lens jargon…

The Nikkor 18-55mm(VR or no VR) is not bad at all, so you might be trying to solve a problem by throwing money at it when it perhaps can be fixed with a small adjustment of your technique, allowing you to make better use of the lens that you own.
Could you post a sample shot in which you find problems that you wish to fix by upgrading your lens?

Edit:
That lens which you linked to exists in versions for all major brands. This offer does not say which particular brand it is for, so i would assume that it's for Sigma's own DSLRs, not for Nikon.
Either way, i don't think that it'd be an improvement over your current lens - both belong to the same price category.

It will fit if it has a Nikon mount. But, in my opinion, it will be a waste of money. The Nikon kits lens is of better quality than the Sigma.
You can get magazine quality photos with the D5000 & kit lens if you take the time to learn how to use them. I should know, I've had photos published with these two.