Xeon is intels enterprise level server chip. "dual core" - dual core can be found in xeons as well as desktop chips such as the core 2 duo and pentium d. There are now dual core xeon chips as well. Dual core is simply put, 1 cpu that functions as two. Before this technology was avaliable you would need two physical chips, but now you could do it with just one.
Intel Xeon CPU's come in both varieties: single and dual-core. A dual-core CPU means that there are two CPU cores on one single CPU. You gain a performance benefit of anywhere from 0% to 80% depending on if the applications you're using know how to use multiple CPU cores.
To give you a real-world example:
I have two servers (one old, one new). The old server has Dual Intel Xeon 3.2 Ghz (single core). The new server has Dual Intel Xeon 3.0 Ghz (dual-core) CPU's. Both had 8GB of RAM.
The old server had a load of 20 at the high points in the day. It couldn't handle the demand hitting it. The new server, which processes the exact same data, has a load of 1.2 at the high points in the day.
If you want more CPU power, you want a dual-core system (if not a dual CPU dual-core system). If you want something that costs less, you need to go with single core systems (such as a single-core Xeon).