I have a dual core machine with FC6 x86_64 which is a dual-boot with Windows XP.

Originally, GRUB would offer the choice of Fedora Core vs. Windows XP at startup. I did yum update recently and then the next time I booted the laptop up, it now has three choices:

Fedora Core (2.6.18-1.2849.fc6)
Fedora Core (2.6.18-1.2798.fc6)
Windows XP

I am assuming that "Fedora Core (2.6.18-1.2849.fc6)" is the one
that I should choose when I want to boot into Fedora.

Is this correct? Thank you.

Sure, unless you want to use the older version. Distros tend to append their own revision number to a kernel. The 2.6.18.1 is the vanilla kernel from Mr. Torvalds and his band of merry (wo)men. Red Hat engineers add patches and append the .2849.fc6. The larger the number, theoretically, the newer the kernel.

The old one is kept as an option under the assumption that your machine was working correctly with it before the upgrade. If the new kernel causes you problems, you can continue using the old one. So your new GRUB menu can be interpreted thusly,

Fedora Core (2.6.18-1.2849.fc6) - new kernel
Fedora Core (2.6.18-1.2798.fc6) - old kernel
Windows XP - spyware? Not sure what this is for.

It's correct,

When Yum updates the kernel, it doesn't remove the old one right away. This means that you'll have the option to boot with a previous "known to work" kernel (the *.2798) with GRUB if anything went wrong.

Are you by any chance running a Core 2 Duo (E6xx) machine, and if so - what motherboard are you using, and is everything working ok?

I'm running a Core 2 duo laptop (with T5500 processor).

I have had nothing but problems getting FC6 to work under x86_64!