The main effects are on image tone and gradation. These can be quite subtle. They can also be quite large, depending on the developer and the paper. Warm tone papers show the most variation, but the subtle effects of different developers on cold tone papers can be very important. Different developers affect different papers in different ways. You really have to experiment. The trick is to match up a paper with one or more developers that work well for your pictures. A good deal of theoretical and practical work has concerned matching the paper curve in a given developer to the curve of a film developed in a certain way (see, for instance, Davis, "Beyond the Zone System"). But a good starting point is to notice the film-devloper/paper-developer combinations that were used for prints you see. Also, listen to what others have to say. Then experiment.
Dektol is a good cold-tone developer, although I prefer the old Defender 54-D. It doesn't do much for warm-tone papers, though. The new coldtone and warmtone developers from Ilford seem to produce outstanding results with most papers.