Quad cores are basically two dual core processors joined together in one cpu package. Looks like one chip, but four processors capable of splitting tasks. I believe that almost all core2 duo and core 2 quads are 64 bit. Last time I looked (over a year) there were a few bin parts in the dual core area that were cheapos that didnt have the 64 bit extensions. About all that really buys you is access to more than 4GB of main memory. There are some other goodies in there to facilitate performance and management of large system loads, but for todays desktop environment its probably not essential unless you're running a big server.
The beauty of multiple cores isnt so much in being able to do twice as much work, its in reducing logjams or blockages. While a single high speed processor can do an awful lot of work but there are many, many operations that applications software perform that require that something finishes before execution of the application can continue. While the processor is waiting on that event, nothing else happens. You'll notice this when you're clicking away or watching something and it looks like nothing is happening.
By rewriting applications to use freer threading models, its possible to eliminate some of these logjams. But thats a lot of work. Adding a second processor core means that when the first one gets stopped, the second one can continue. Massive applications like Microsoft Office use some complex threading constructions that can stop a single processor machine an awful lot.
Some applications can be very processor intensive or interrupt the processing load a lot. Those will make your computer seem very slow when working in the foreground doing your web browsing and email. Backups, video conversions, dvd burning, virus scanning and other such products can soak up an entire processor, but a second/third/fourth available allows you to run your foreground apps independent of that background load.
At this point in time, many apps are made to work on a single processor, but that doesnt stop a multi core machine from breaking up a total system load across multiple cores. Most modern OS's are running 50-200 processes at any given times with generally 2-4 being very active at any given time.
As time goes on, more applications will be threaded more freely which will allow one application to spread itself over all 2 or all 4 cores, so it'll better facilitate multithreading as well as multitasking.
Some good reading here on the various ways app threading works.
Processes, Threads, and Apartments
Honestly, at this time quad cores are a little overkill for todays home desktop apps unless you have enormous computational requirements. But 3-4 years from now they might look positively essential.
When you can get four cores that each have more horsepower than the very best cpu offered just 2-3 years ago, the futureproofing is pretty solid. At these prices you're not overpaying for the pleasure either.