For the most part, colleges require a baseline of at least two years of high school math, including Algebra II at minimum. This baseline is somewhat unclear, however, as different high schools have different standards for what two years of math and algebra entail.
It is important to check the specific math requirements and recommendations for each of the colleges you are applying to in order to make absolutely sure you meet their requirements for application.
In general, high school students should strive to take Algebra I, Algebra II, Calculus and Trigonometry during their high school career. This class progression follows the four years equals four credits formula that most high schools adhere to.
At minimum, students should complete Algebra I and Algebra II, although this may limit the college options when it comes time to apply. If the option is available to take more challenging math classes within the high school, students should take advantage of it as well.
For students applying to a math-intensive college or intending to go into a math-related program, math classes above and beyond the typical high school offerings may be required. The Advanced Placement (AP) program is an excellent way to prepare for future math study, and test scores from the AP exam can play a large part in acceptance to a particular college.
If your high school offers the opportunity to take math classes at a college level, that should also be taken advantage of to prepare for college applications.
If you did not have the opportunity to take the level of math required by the college you are applying to, you may still have options. Many colleges offer the opportunity to "test out" of a math requirement by administering a one-time test that covers the basics of their math requirement and proves to the school that you are adequately prepared.
Other schools may offer a remedial math course for freshman that will get you to the level desired by the school during your first year of college.