Difference betwee a palm, fixed base and plunge router?
Routers have collets (the part that takes the bits) that range from 1/4 inch to 1/2 inch, with the (very rare) router that can take 3/8 inch diameter bits. If you are planning to do heavy work, such as cutting a 3/4 inch groove in hardwood, you are going to want to use 1/2 inch diameter bit, not a 1/2 inch diameter bit. If that is the type of work you are planning, you should purchase a router than can take 1/2 inch diameter bits. Typically such a router will come with an insert so it can take 1/4 inch diameter bits as well, however the small palm and trim routers often can only take 1/4 inch diameter bits.
Huge industrial grade routers can often take 3/4 inch diameter bits, but it is most unlikely you will ever use such a beast. Personally I have three routers, one is a 3 HP plunge router which does the vast majority of my work. One is a small fixed base router, very light, takes 1/4 inch bits, which I use for trim work. One is an old Sears Craftsman fixed base router, which I do not use any more, since the base refuses to lock firmly in position.
Couple things are crucial with a router. First, the bits need to stay rigidly locked in position, which is a function of how well the collet is made. Cheap routers often have poorly made collets, and the bits can loosen while your are working, which is incredibly dangerous. Avoid such a router. Second, the router base needs to lock firmly in position, which is a function of the quality of the base locking mechanism. Cheap routers often have weak or poorly machined locking mechanisms, so the base can slide up during the cut, which can ruin the cut. Make sure the router you buy has a good base locking mechanism.
There are things you can do with a plunge router than you simply cannot do with a fixed base router, typically starting a cut in the middle of the piece rather than having to start from the edge. This is often very useful. The fixed router is generally lighter than the plunge router, but for my money, I would get a plunge router due to versatility, if you are only going to get one.