Sopranino - as seen being played in the Ukulele Orchestra of Great Britain www.ukuleleorchestra.com this is a tiny showpiece for players who have tiny fingers, are committed to making playing as difficult as possible for themselves or want to be able to fit the uke in their back pocket or handbag.
Soprano - the most often played and least expensive version around. It is still pretty small but even big-handed tradesmen and truck drivers have been known to navigate the fretboard and strings successfully. The most common tuning for the strings is - from your chin to your feet - G C E A.
It's called re-entrant tuning because unlike a guitar, where you start with the lowest note and move up in pitch as you strum down, the uke starts high (G) then low (C) and upward from there (E & A).
Concert - With a name that sounds like you're duty bound to perform while playing, the concert uke only differs from the soprano in that it is slightly bigger. Might suit your fingers better, you might like the sound of it or you want to get something different from or more expensive than your neighbour.
Concert ukes sometimes have a pickup built into them, which means you can use a lead to plug your instrument into an amplifier and get a louder sound. Tuning is G C E A.
Tenor - Blokes particularly seem to like tenor ukes as their larger size can more easily accommodate big fingers and hands. Like concerts, they also sometimes have a pickup,
standard G C E A tuning and can also be fitted up with a low G string. Contrary to reports, this isn't saggy underwear, it's a wound metal or thicker nylon string that is an octave lower in pitch
than the standard high G. It gives the instrument a more guitar-like sound and can be used in conjunction with the C as a bass string when plucking.
Baritone - Now we're moving out of standard uke territory. The baritone ukulele is tuned the same as the four highest pitch notes on a guitar (D G B E from your chin to your feet), but it's still a uke and can be very beautiful and mellow sounding. The body of a baritone is generally even larger than a tenor, but still nowhere near the size of even a travel guitar.
Taropatch - Otherwise known as an 8 string ukulele, these are also tenor sized or larger with G C E A tuning and each string doubled and an octave apart, very much like the ukulele version of a 12 string guitar. This gives them a distrinctive and louder sound than a standard 4 string uke.
6-string - Not a guitar, it has G C E A tuning with the C and A strings doubled, an octave apart. Usually tenor sized, these ukes again have their own sound.
Banjolele - Ukulele tuning (G C E A) with the body of a very small banjo. They sound a bit like a banjo and the skin on the body makes them very much louder than a ukulele, sometimes drawing annoyed looks and derision from uke players if the banjolele is played in a group! They're great for making anything sound like bluegrass, however.
Others - There are also metal resonator ukes and probably a host of other interesting variations I haven't yet encountered, so feel free to fill me in! firstname.lastname@example.org