To the musically minded, these are all different pitches on the musical scale. A baritone is lower in pitch than tenor, alto is higher than tenor, soprano is higher than alto. The saxes corresponding to these sizes are essentially the same instrument, but they do look very different. A baritone sax is a big instrument! Soprano, on the other hand, doesn't really look like a sax at all when you first see one. They look more like metal clarinets, when they are the common straight versions. Alto and tenor saxes are by far the most popular. You can identify the saxes using this comparative process:
Is the crook (the tube leading to the mouthpiece) straight, hump-backed, curly or indistinguishable from the body of the sax? The latter case describes a soprano, while a straight crook is the shape for an alto. A hump-backed crook is a tenor, while the curly crook is a baritone. The size of the instrument is also quite telling, if you're not sure. The tenor is substantially larger than alto, for example. An alto is generally played in front of the body, while a tenor hangs by your side. I was recently informed that some military bands have the alto played by the side, and accordingly some alto saxes have been designed slightly differently. You may like to ask firstname.lastname@example.org for more information on this point.
Why the different sizes? Well, even though they have the same fingerings and can play notes from low Bb to high F#, the registers are very different, and the tone is noticeably altered from one to the next. Remember that saxophones are transposing instruments: soprano and tenor are Bb instruments, with soprano being one octave higher than tenor. Baritone and alto are both Eb instruments. This means that if you play a C on a saxophone, the note which a pianist would play to match your note would either be a Bb or Eb, depending on the sax. This is quite a common thing in orchestras: clarinets, trumpets, horns, double-reed instruments and the likes all have to have music written appropriately to make them sound the correct notes. The background to this is historical, but it does for instance mean that you can buy a tutorial book written for `saxophone' and it will be applicable to all of the different sizes. Accompanying instruments will have to transpose if you don't, however.
The sounds you can get from these saxes, and their capabilities are intrinsically different - choosing which to play is difficult for a novice. To some extent you might need to think about your lips and embouchure (lip muscles) and then choose: Tenor saxophone is reputed to be the easiest to start on, but by no means the easiest to master. It has a versatile and rich tone when played skillfully, and needs a firm embouchure suited to most mouths. Altos require more lip control, having a more piercing but very expressive tone, again accessible to most mouths, while baritones are a little slower because of their size and mechanisms. A loose embouchure can make these a good choice if you're not able to get to grips with the smaller saxes. They can be played solo or as a bass instrument.
You can, generally, start sax on any of these instruments, and eventually they become interchangeable with practise. You are not limiting yourself by playing only one sax to begin with (remember, the fingerings are the same - it's the mouthpiece which will need practise when changing). You may be happy with just one size of sax, moreover. Many players would agree with you. The only sax which cannot be taken up by any right-minded beginner is the soprano: these are noticeably more difficult to play, requiring much more airstream control. In general, an alto is a better foundation at first if soprano is what you aspire to. Sopranos are a prize in themselves, though, and have a melodic tone which is hard to match with other saxes.
One last thing; more types of sax do exist. Sopranino is the smallest sax (and quite unusual) while bass saxes are available in some places. They are huge, but if it's not enough for you to own a bass, try looking for a contra-bass...very few remain in existence, but contra-bass needs elephant lungs to play and an 18-wheeler for transport. :-) Also, there are C-Melody or C-Tenor saxes, which are not transposing instruments. They are not in common use and can be difficult to get parts and reeds for. If you need a C-Melody mouthpiece, contact John Myatt woodwind as detailed under part 2.3.
I've recently come into possession of some photographs of an original Adolfe Sax saxophone thanks to a mysterious person from France.
A chart of saxophone ranges (from Michel van Assendelft)