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).