i386 refers to the 32-bit edition and amd64 (or x86_64) refers to the 64-bit edition for Intel and AMD processors.

Wikipedia's i386 entry:

The Intel 80386, also known as the i386, or just 386, was a 32-bit microprocessor introduced by Intel in 1985... This is termed x86, IA-32, or the i386-architecture, depending on context.

Wikipedia's x86_64 entry:

x86-64 is an extension of the x86 instruction set. It supports vastly larger virtual and physical address spaces than are possible on x86, thereby allowing programmers to conveniently work with much larger data sets... After launching the architecture under the "x86-64" name, AMD renamed it AMD64... x86-64 is still used by many in the industry as a vendor-neutral term, while others, notably Sun Microsystems (now Oracle Corporation) and Microsoft, use x64.

Even if you have an intel CPU, you should use AMD64 to install 64-bit on your computer (it uses the same instruction sets).

I highly recommend using it. For the most part you will not notice a difference but for large workloads (such as video editing, gaming, etc), the computer will perform faster (the computer has the ability to calculate 2+2+2=6 instead of having to do 2+2=4+2=6 in an example). In the Windows world, a 32-bit OS will not let you use more than 3.5 Gigs of RAM on your computer (even if you have 8!). You'd need to use a 64-bit OS in order to fully use all RAM. For Linux, however, there's no such limit (thanks, Uri).

Regardless, the world has shifted from 32 bit and it's only there to support older machines that are incapable of running 64 bit.