A research statement lays out your vision of your research, including both your accomplishments to date and your plans for the future. It is a required part of many applications — for graduate school, for fellowships, for jobs (especially academic jobs), for awards, etc.
This is a short collection of advice regarding some mistakes I have seen people make in their statements. It is incomplete at present, but still useful if it helps you avoid those common but important blunders.
Write your statement in the first person.
Your statement will likely discuss multiple research projects or results. Readers are unlikely to remember each of the projects in detail (though they may remember that you are broad, which is good). It will be easiest for readers to remember your work, and equally importantly to place it in context as they read your statement, if you can show how the projects are related. They might be in the same research subarea, or use related techniques, or stem from your own experiences, etc. Find a way to tie them together, which will make readers more likely to understand them. Furthermore, readers will be more likely to believe that you may continue to produce similar results in the future, if they see a pattern. After starting out with an overview (which might describe the general topic and its importance), tie the description of each result or project into that overview.
Somewhere — maybe at the beginning, though more likely at the end — you need to bring the statement to life and let your personality show through. Permit yourself to gush about how much you enjoy research: show that you really care about this and why. For example, you might state how important it is to you and how it has changed the way you think about your research area computing. Another way to address this (not incompatible with the others) is to say why you got involved in research at all; that might go at the beginning. If readers believe that you are enthusiastic about the topic, they will be more likely to believe that you are doing a great job and will be more favorably inclined toward you in general.