I'm not going to pretend to know every possible fellowship that exists for psychology research. In general, the main point that I want to emphasize in this post is to use the Fellowship and Research Office's website in order to learn about fellowships.
Mostly, what we are talking about here are opportunities during a gap year(s). Fellowships (and scholarships) like the Rhodes, Fulbright, Marshall, or Gates (among many others) are research opportunities for study in (usually) a foreign country for 1-2 years before one goes to graduate school. Sometimes the fellowship allows you pick up a master's along the way.
I want to say two things about these fellowships. The first is not about the fellowships themselves, but about a concept that begins to creep in here when students think about fellowships. There are a lot of students who think - wow, Rhodes Scholarship or Marshall Fellowship - those are really fancy. That's not for me. That's for someone different.
And maybe that's true. But maybe it's not. Yes, some of these fellowships are based on grades, and you can be a good judge of whether you have had good grades during your Brown career. But the main idea is that there are a lot of these fellowships - including many that you have never heard of that present unique opportunities for all types of students. You don't know until you look around for them.
The second thing that I want to mention is not a fellowship, but a postbac program called the IRTA. It's a program done through NIH, where students apply in their senior year of college to go and work at the NIH for (usually) 2 years. Here is the website, and it's something that is super interesting to consider, particularly if you are interested in applying to a clinical Ph.D. program, but not wanting to do so right away.
By the way, it's a coincidence that IRTA sounds like UTRA. I get them confused in my head often.
Update: There's another fellowship that gets advertised most years. It's the Sara Sparrow Fellowship in Clinical Neuroscience of Autism at Yale University. It's a 2-year postbac fellowship for students interested in Autism. I'm writing this on Oct 27, 2023, and the call for 2024 isnt out yet. But here is the call for 2023.
Another update: I spoke with some former students, all of whom were recipients of fellowships. One of them, who is now a Full Professor and Fancy Dean at another university, mentioned to me that I left something out of this post - which is that you have to start early. These fellowship applications take a long time to write (which is true), and her recommendation was that students have to start putting your application together in the Spring of your Junior year. I can endorse that. So, if you are thinking about these, it's good to start early. Spring of Junior year *might* be too early, but definitely the summer between junior and senior year, which incidentally, is also a good time to start planning for graduate school if you are so inclined.