CACHE, CONT.
The analogy I heard to help keep cache straight is thinking about a library. The harddrive is the stacks. Each time you want a book, the librarian had to go to the stacks to grab it. But if the librarian kept the top 20 most commonly checked out books at the front desk, those operations would be faster. Think of the front desk as the cache. The larger the front desk (cache), the more books (operations) can be stored for quick retrieval. L1, L2, and L3 are levels of cache, and would be analogous to, say, front desk, desk behind front desk, and desk behind desk behind front desk - three levels before you have to go to the stacks.
64 BIT vs 32 BIT
64-bit and 32-bit are based on instructions; the only think I know about it is that 32-bit operating systems can only use a little less than 4GB or RAM, because it simply does not have a big enough address space to specify anything larger than 4GB. 64-bit on the other hand can go beyond 8GB ram. My guess is that a CPU needs to be compatible with the 64-bit architecture in order to utilize a 64-bit operating system and thus larger amounts of RAM.
HOW MANY CORES THEY HAVE
For your single core vs. dual core, check out this link:
Common Noob Myths - Computer Forums
It answers your question and potentially others you might have in the back of your mind. I know I've made several of these assumptions/mistakes.