My impression is that REU's (even at not so elite schools) are very competitive to get into. I applied to 12 last year and only go in to 2, including rejections from several places where I could very likely get into their Phd program. I know people (with comparable qualifications to me) that didn't get into any.
As for what to include in a cover letter/personal statement, I wrote primarily about research I had done as well as significant mathematics I had seen that might not be evident from my transcripts or CV (e.g. independent studies/seminars I'd participated in). There was a fair amount of mathematical terminology/notation, but of course it was very informal, only attempting to communicate the conceptual meat of the mathematics I was discussing.
One of my letter writers was a professor I had done extensive research with in his area of specialty, and the program I attended was at the school where he did his Phd. If you have a glowing letter from someone who has connections at the program, it probably doesn't matter that much what you write, provided it conveys competence and enthusiasm.