You're right that there is lots of variety. Towards the end of 2nd year you have to apply to homebases. These aren't the projects, but the areas in which you can work. As well as your research project there will also be exams/coursework unique to each homebase. These should also factor into your decision making, as there are huge differences. My homebase, psychiatry, had no exams but we had to do 5000 words of coursework on top of the project. This compares to exams in early November for clinical sciences or in late December for biomedical sciences.
So you rank all of the homebases and allocations are made using a best fit model. Most people get something they at least don't mind, but a few prime specimens will always forget to fill in the form and get sent to molecular pathology. Then, after 2nd year exams you go to your homebase and get allocated a project. Each place will do it slightly differently, but most will essentially ask you to rank all of the available projects and they will do their best to avoid giving too many people their 10th choice. I got my 3rd choice.
The types of projects, and the level of input that you can have into defining your research parameters vary wildly. More sciency homebases will have predominantly lab projects, with literature reviews thrown in for some. Clinical sciences have patient data studies, audits, surveys of clinicians and some device/treatment testing. One that seems to crop up every year involves getting lots of people to cannulate chickens under ultrasound guidance.
Psych seems to be one of the most varied areas to work in; it's a very small homebase (about 18 people) and the types of projects on offer seem a little more heterogenous than lab or lit review. The research interests of the department mean that ADHD and dementia are the most common areas, and people do everything from qualitative studies of art therapy in Alzheimer's disease to gaze direction based studies of attention and reaction times in ADHD and schizophrenia. Other people did surveys in schools or even high security hospitals.
My project was a Cochrane systematic review of a particular antipsychotic drug used in schizophrenia. This was pretty different to the rest of the homebase, and as far as I know I was the only person in the year as a whole working with the Cochrane collaboration. There were quite a few issues with the project and some of the support I was able to get, but if you're an EBM geek like me, the chance to be involved with something as important as Cochrane was too good to miss. The downside is that I am still working on the bloody thing to get it published.