The Mathematics Department has hosted a wide diversity of Undergraduate Research Opportunities Program (UROP) experiences. Every year the Department hosts around 70 different UROP projects.
In some cases, relevant UROP projects will be advertised in the UROP job listings, and it is worth checking these periodically. In most cases, however, UROP projects are developed in consultation with you after you contact an advisor and find an area of mutual interest, sometimes beginning with a directed reading project to learn background material and narrow your focus. UROP projects of a mathematical nature in other departments can also be very helpful in preparing you for further research and careers in mathematics, so don't hesitate to consider such opportunities as well. Over the last few years, more than 80% of active math faculty have supervised one or more UROPs.
Typically, you should begin by browsing the MIT mathematics research listings and identify an area of interest to you, and then contact one or two potential advisors to talk about research in their area. Instructors of courses you have taken can also be useful points of contact. At this stage, remember that a UROP project may not have been formulated yet — your first goal is to get to know the advisor and their research area, as well as to learn what courses might help further your interest in that area (see also the math roadmaps). After this discussion, if you are interested in working with that advisor (regardless of your previous experience in that area!), you should feel free to ask them whether a UROP might be possible. Even if the advisor can't identify a good problem at that time (or is already advising other UROPs), they should be able to introduce you to other potential advisors working in related areas at MIT. (Often, there will be many potential advisors you may not be aware of, including postdoctoral researchers, visiting scholars, and faculty in other departments at MIT, and you may also be co-supervised by a graduate-student mentor.)
Persistence and enthusiasm are important. Opportunities for UROPs will increase as you take more courses and learn the vocabularies of different fields of mathematics, but sophomore-level courses should be sufficient to find a project. Finding a UROP in mathematics will be more difficult for students who have only taken first-year calculus, but there are often still possibilities for students with other relevant skills, such as programming experience, or for developing educational materials. Realize also that the prerequisites vary by project — the same advisor who wasn't a good fit for you one semester might have a different project that works for you at a later time.
For information, visit the MIT UROP website.