The Unix kernel acts as a mediator for your programs and your hardware. First, it does (or arranges for) the memory management for all of the running programs (processes), and makes sure that they all get a fair (or unfair, if you please) share of the processor's cycles. In addition, it provides a nice, fairly portable interface for programs to talk to your hardware.
There is certainly more to the kernel's operation than this, but these basic functions are the most important to know.
Newer kernels generally offer the ability to talk to more types of hardware (that is, they have more device drivers), they can have better process management, they can run faster than the older versions, they could be more stable than the older versions, and they fix silly bugs in the older versions. Most people upgrade kernels because they want the device drivers and the bug fixes.
See the Hardware-HOWTO. Alternatively, you can look at the `config.in' file in the linux source, or just find out when you try `make config'. This shows you all hardware supported by the standard kernel distribution, but not everything that linux supports; many common device drivers (such as the PCMCIA drivers and some tape drivers) are loadable modules maintained and distributed separately.
Linus recommends a version of gcc in the README file included with the linux source. If you don't have this version, the documentation in the recommended version of gcc should tell you if you need to upgrade your libc. This is not a difficult procedure, but it is important to follow the instructions.
These are pieces of kernel code which are not linked (included) directly in the kernel. One compiles them separately, and can insert and remove them into the running kernel at almost any time. Due to its flexibility, this is now the preferred way to code certain kernel features. Many popular device drivers, such as the PCMCIA drivers and the QIC-80/40 tape driver, are loadable modules.
It depends on your particular system configuration. First, the compressed linux source is nearly 14 megabytes large at version 2.2.9. Many sites keep this even after unpacking. Uncompressed and built with a moderate configuration, it takes up another 67 MB.
With newer machines, the compilation takes dramatically less time than older ones; an AMD K6-2/300 with a fast disk can do a 2.2.x kernel in about four minutes. As for old Pentiums, 486s, and 386s, if you plan to compile one, be prepared to wait, possibly hours, days..
If this troubles you, and you happen to have a faster machine around to compile on, you can build on the fast machines (assuming you give it the right parameters, that your ulilities are up-to-date, and so on), and then transfer the kernel image to the slower machine.