Technically x264 is not the actual codec. The codec is H264 which is a MPEG-4 codec. Divx and Xvid and also MPEG4 codecs but H264 is from a differnet part. x264 is the software library that provides the ability to encode H264.
If your just interested in downloading tv shows and movies the technicalities are not so important and x264 and H264 are interchangable. The x264 codec is used for low quality podcasting and high quality video downloads.
MPEG 4 is a standard with many parts and there are substantial differences between them. ASP versions like Xvid and Divx and AVC versions that x264 handles are well more than you need to cover for just downloading and playing movies and tv shows that you download.