Your system is 64-bit capable, and having installed 10.8 means you are running in full 64-bit mode at all times. You can still run 32-bit applications if needed, but most new ones should be developed with 64-bit code.
The difference is primiarily in the size of numbers that can be addressed. A bit is akin to a transistor gate being on or off (one or zero). The number of transistors that represents a number is the "bitness" of that computer. For example, a 1-bit computer can only count up to 1 since it has the options of 0 and 1 (off or on). A 2-bit computer uses two transistors to represent a number, so it can count up to four (2^2) in the following patterns:
00 = 0
01 = 1
10 = 2
11 = 3
(zero through three is four individual numbers to the computer)
A three-bit computer has double the amount of patterns, so it can count up to eight (2^3) individual numbers:
As you add more bits to what constitutes a number in a computer, you add more number possibilities that the computer can handle. With 32-bit computers the system could manage numbers as large as 2^32, which is only about 4.3 billion. This is a large number, but given the vastly expanding computing demands these days it is actually a minuscule number. For instance, the 4.3 billion limit means a 32-bit computer cannot assign more than 4GB (four billion bytes) of RAM to a program.
Since these days there is a lot of demand for RAM from some programs (movie editing, photo editing, etc.), then the 32-bit computers run into these problems. Using a 64-bit computer means you can easily assign and manage 4GB, 8GB, 16GB...etc up to 2^64GB (trillions and trillions) of RAM without running into such roadblocks.
64-bit computing enables other capabilities and features beyond RAM management, but RAM addressing is the primary benefit seen. Luckily, all these benefits and capabilities are primarily for the programmer, and for the vast majority of people the 32-bit or 64-bit difference will ultimately not amount to much (except allow advancements such as prettier interfaces and slicker experiences as programmers make more and more use of these technologies).