While there are many REUs, there are also many, many students every year who apply. As a result, getting into an REU has become very competitive. Below is some advice that might help you as you apply.
Apply to several different REU sites. As good as you might be, do not count on this to guarantee your acceptance at your "dream" site. You need to give yourself multiple opportunities by applying to multiple sites. At the same time, make sure that you meet the requirements for each site to which you apply. A good rule of thumb is to apply to 6-8 REU sites.
Do not limit yourself to only certain topics or certain geographical areas. Some people will only apply to sites which fits only their current interests, or are located in a particular geographical area; do not limit yourself. Broaden your horizons, give yourself opportunities for new experiences, you might even find you liked it more than what you had originally wanted.
Follow the directions. Each site has its own set of materials that applicants are required to turn in. Make sure that you follow the directions and give them what is requested. One of the easiest and fastest ways to disqualify yourself from an REU is to not follow the directions. If something is unclear, contact the REU site (they are happy to clarify things).
Carefully read all your application materials, and have someone else read them as well. While these are not REUs in English, the essays that are submitted are used to give a sense of the applicants ability to communicate. Make sure that you avoid spelling mistakes, use correct grammer, and have something interesting to say. On the other hand, don't try and be too creative! Treat this as an application for a job (it is!), be professional.
Tailor each application to specifically address the REU for which you are applying. It is certainly easy to write one letter and just change the name of the program to which you are applying; but it is also easy for people reading your application to recognize this. Take some time and get to know more about the REU, about the topics that will be covered, any special groups the REU is targeting that applies to you (e.g., minorities, women, first-generation college), and about the institution where the REU will be hosted. Use this information to personalize your letter and/or essays to the REU. Be as specific as you can in your statements.
Don't sell yourself short. Some people are modest, and don't like to talk about themselves. This is wrong! Talk about the great things that you have done and what makes you someone who is a good fit for the program to which you are applying. When appropriate talk about what your long-term goals are, many REU sites target students with certain career goals (e.g., teaching or graduate school). (Of course, be honest. If you lie on your application, this can cause problems for you down the road.)
Think carefully about your letter writers, and make sure that they know you well. One of the key components of the application are the letters of recommendations. You will need several. Start thinking about who can write letters for you, ideally they should be someone who has firsthand experience of your mathematical abilities through either a classroom setting (the higher level of math class, the better; also the more often they had you in class, the better) or through conducting research (if you have previously done an REU, then one of the letters should be from your former REU mentor/advisor). When asking someone to write a letter of recommendation, ask if they can write a "good" letter of recommendation (in the sense that they are enthusiastic supporters of you), if not go find someone else.
If possible, get more senior people to write letters. A letter from a Professor is taken more seriously than a letter from a lecturer, which in turn is taken more seriously than a letter from a graduate TA, which in turn is taken more seriously than a letter from a high school teacher, which in turn is taken more seriously than a letter from your Mom (love you Mom!).
Give your letter writers sufficient time (at least one month if possible before the first deadline) and give them additional information (e.g., why you want to do an REU, what your long term plans are, relevant experiences, transcript, etc.). In other words, give them the tools they need to write that good letter for you. Where appropriate, ask letter writers to address your ability to work in groups (this is an important part of the REU experience which can not be determined by transcripts). Also note that letter writers can be a useful way to address "problems", for example if there are issues with grades.
Start your application process early. This process is not hard, but it does take time to do it well. Ideally you should start preparing at the end of the Fall semester for next summer. Keep in mind that some sites might not even start advertising until March, so make sure to keep informed about places that are running REU sites.
Have a backup plan. If you don't get into an REU, have a plan on how else you will spend your summer. There are usually other options, for example summer courses (if your school does not offer them, a nearby institution might and then you can transfer credits); research experience with faculty at your own school; internships; and so on. Remember that REUs are not a requirement for graduate school, there are also other opportunities that can be just as rewarding and helpful.