MBR is Master Boot Record - an area of 512 bytes on the hard disk:
(wow - someone actually entered good info on the wikipedia)
All 'DOS' partitions have a similar set of 512 bytes set aside containing the partition record (which also has a small area for boot code, just like the MBR).
A 'rescue disk' is some disk you can boot to and which can help you fix problems with the installed system; many installation disks have a 'rescue' mode although they're often a hassle to use. A 'Live CD' is a CD you can boot and run from - it is often more convenient to boot from one of these when fixing a system because you have far more tools available than the typical 'rescue' disk.
You don't tell the installer to boot to MBR, you tell it to install the Bootloader in the MBR (as opposed to the boot section of the partition record). If you look at the wikipedia article, you'll see that there's very little space for the bootloader in the MBR or partition record, so what actually happens is that the "first stage bootloader" is stored there - a very tiny program whose only purpose is to load and start the "second stage bootloader".
Since ancient BIOS has a very low limit on how much it can access on a disk, and the first stage bootloader needs BIOS to read in the second stage (and the second stage needs BIOS to read the kernel and the 'initrd' image), for machines with an ancient BIOS you need a small partition at the start of the disk which will hold the second stage bootloader, kernel, and initrd file. The kernel itself does not have such a low limit, so once the kernel starts and is fully initialized, the system has access to the rest of the disk. Also, when the kernel is initialized, no one uses BIOS calls anymore - the kernel takes care of all read/write requests. In fact, you can't run any BIOS instructions anymore (well, not without the kernel's help and at great inconvenience).