Basically a duplicate of this question. It comes up a lot, and the remove category base functionality probably should be an option in WordPress to be honest.
Here was my answer from there:
No Category Base plugin to remove the /category/ bit from the URL.
🙂
If you have a page with the same name as a category, the page will be shown instead the category archive, but the URLs of posts won’t be affected. At least, that used to be the case. I think I heard news of a recent change in the way WP calculates permalinks (for the better).