No, you don't have to have a kekkei genkai to have two elements. According to Kakashi's explanation, most people start off with one element. Jonin and higher can control two or even three, so it appears that you gain the capability to affiliate with more elements as you train more.
Kekkei genkai =
the ability to combine two elements into another oneEx: Haku can control water and wind. Other regular ninja can control water and wind, but since Haku has a kekkei genkai, he is also able to make ice. Keep in mind that this is only one kind of kekkei genkai; there are others like dojutsu (eye techniques, Sharingan and Byakugan).
I think it's possible to do any element jutsu but if you're not affiliated with that element, I guess it's less effective?
Explanation from Kakashi himself:
http://groups.msn.com/NarutoMangaReturns/rvolume35.msnw?action=ShowPhoto&PhotoID=31128((Chapter 315, he starts off with talking about training with the tajuu kagebunshin first, but it goes into elements in a few pages))