Plasma is blood minus the blood cells (red blood cells, white blood cells and platelets). It is therefore largely water, but contains numerous other things such as proteins (especially albumin), glucose, lipoproteins, amino acids, hormones, ions (e.g. sodium, potassium, etc.) and carbon dioxide, to name but a few.
Serum is the name we give to plasma that has had the clotting factors removed. You'd usually achieve this by allowing the sample to clot first, thus using up the coagulation factors.
Since allowing the blood to clot also traps all the blood cells, most chemistry tests of the blood are serum tests, not plasma tests. It's just a lot easier to get at - once the blood has clotted, the remaining fluid will be almost entirely serum. (You obviously want to measure the plasma for things like clotting factors, though, so these samples have an anticoagulant in them to keep the blood from clotting.)