The original music compression codec, and some still maintain the best, is the most widely supported of all the codecs available, though there's plenty of older hardware that doesn't support VBR features. There are loads of different encoders you can use, but the most popular and the one deemed the best is LAME. I've used this (version 3.97, the latest stable non-beta version) for the listening tests you'll see later in this feature.
But while it is still a very good codec, it does have its limitations. Though highly tweakable and customisable from the command line and capable at medium quality levels, MP3 isn't very good at higher compression ratios (below 128kbps CBR) and doesn't go higher than 320kbps, as previously mentioned.
The latter is important, not simply from the perspective that music can't be compressed at super high quality, but because, in VBR mode that limits the maximum amount of storage space allocated per musical 'frame' to 320kbps as well (compressed music is represented by slivers of data; each sliver, representing a fraction of a second of audio information, is called a frame). And in some circumstances, that 320kb frame isn't big enough.
Its joint stereo mode is not as sophisticated as other codecs, how it deals with audio signals above 16kHz is not ideal and the way it splits the signal into sub-bands for the psychoacoustic model to do its work is over complicated and lossy. The more modern codecs use a single, lossless, transform to split the signal into sub-bands. The details behind the latter are quite complicated, however. If you want to read more about about them, visit MP3 Tech and sink yourself into the maths.
The final limitation of MP3 technology is nothing to do with music or sound quality or file sizes, but money. The technology patents behind MP3 are owned by a company called Fraunhofer-Gesellshaft, who originally developed the technology back in 1991. And since 1996 all developers of MP3 encoders and those distributing MP3 via streaming (ie for Internet radio) have had to pay a licensing fee to the company for the privilege.
For this reason, development is likely to be slower than that of free and open encoders such as Ogg Vorbis, and improvements slower to emerge. The popularity of the codec may also wane as other more modern and free technologies begin to take over.