How important are standardized test scores compared to other pieces of the application?
Standardized tests are only one piece of the application puzzle. The original intent was to create a benchmark so that colleges could feel assured that a student who goes to a "name" private school and a student from a large urban public high school could display knowledge "equally". It is not a perfect system, as we all realize--some people receive tutoring to perform better, some people receive accommodations that others may view as an unfair advantage (this is not true, but that is the subject of another question!), and some people who are at the top of their class are just not good at taking these types of tests, even though everyone knows they are perfectly capable students--but it does work as part of a process to find out some specific information about a potential candidate.
What is very important to understand, however, is that because this is only one piece of the puzzle, you want to insure that you have also covered your other bases. If you get high SAT scores, for example, and have poor class grades, write a poor essay, do not take part in any extracurricular activities, and do not interview well, you are not helping yourself. On the other hand, if you get "only" average SAT scores but you have great class grades, a wonderful personal essay and recommendations, and agree to an interview, the standardized scores will become less important than other factors. Also, if your total GPA is somewhat depressed due to poor grades in your first year of high school, but you've performed better AND gotten good SAT scores, this can help you as well.