Which version to download

Use amd64, which Linux distribution will you be installing?

Ah, but Debian has Stable, Squeeze and Sid. So, the question is still valid.

Both i386 and amd64 would work, but you probably want the amd64-version for performance reasons.

The i386-version is compiled for 32-bit processors, and should work on most derivatives of the processor Intel 80386 (which was initially released in 1985), including almost all modern processors from Intel and AMD. Some closed-source software for Linux has traditionally only been available for this architecture, but due to the widespread adoption of the amd64-architecture in recent years, this should not be a problem anymore.

The amd64-version will not only work, it will also enable you to use the full 64-bit capabilities of your processor. Essentially, this means that your computer can work with more data at a time (64-bit registers instead of 32-bit registers) and make use of more memory (32-bit processors can only address a maximum of 4 GB of RAM). Since compatibility with old processors is not required by the amd64-version, I believe this version also comes out-of-the-box with more optimization for other capabilities of modern processors (e.g. SSE-instructions).