A 64-bit processor is one which handles (or can handle) 64 bits of data in a single operation (for example addition of two registers each containing 64-bit integers, or accessing a memory location via a 64-bit pointer).
A 64-bit operating system is one which is compiled to take advantage of the 64-bit instructions provided by a 64-bit processor and as such will not work on a processor that does not support these instructions. One advantage of 64-bit Windows for example is that it can take advantage of more physical RAM than the 32-bit version of Windows.
Many mainstream 64-bit processors also have 32-bit instructions so that they are able to run either a 32-bit or a 64-bit operating system.