In order to use a 64-bit client of WoW, you need to have a 64-bit operating system on your computer. The main difference between 32- and 64-bit operating systems (usually Windows) is that one deals with information in a more efficient way. Information in a 32-bit operating system is dealt with in pieces of information that are 32 bits wide, while 64-bit operating systems deal with information that is 64 bits wide -- double the alternative. With this larger bit structure comes better efficiency and the ability to use or "see" more RAM in your computer. 32-bit systems can only use 4 gigabytes of RAM and your 32-bit WoW client can only use 2 gigabytes, whereas the amount a 64-bit system can use is almost unlimited.
While we don't fully know exactly what improvements to the game will come with the 64-bit client, we can assume that more efficient processing power and the ability to stack more RAM for WoW to use might give some players a big frame rate boost and better performance. Lifehacker has an old article about how to tell if you have a 32- or 64-bit operation system.