Photo: Three "vintage" styled ukes by Mele; mahogany soprano (standard), koa "pineapple" soprano and  mahogany concert. 

    From 1879 up into the mid 1920's, there was only one size of ukulele, the soprano. Now there are at least four. See the following Ebay guide for the distinctions between, and dimensions of, the various sizes and how to tell which size you've got;
http://reviews.ebay.com/Ukulele-Soprano-Tenor-Baritone-Which-Uke-is-Which_W0QQugidZ10000000001575379.

     Also, there are variations on several of those sizes incorporating 5, 6, 8 or even 10 or more strings. 6 and 8 string tenor ukes are fairly common. The 6-string tenor uke is actually more popular than the 8-string, even though it hasn't been around nearly as long. (I'm told that the first 6-string ukes were built by Kamaka in the late 60's or early 70's.) The 6-string tenor has one G, two C's (in octaves), one E and 2 "A's" (in either octaves or unison), while the 8 string doubles the G and C in octaves and the E and A in unison (much like a 12-string guitar>  Probably the main reason the 6-string has become more popular than the 8-string is that it is more versatile, lending itself to either picking or strumming styles, while the 8-string is more conducive to strumming.

Early 8-string ukes were small, somewhere between standard and concert size, whereas modern 8-strings are usually tenors. The 8-string was once known as a "taropatch fiddle", sometimes shortened to just "taropatch", or  a Lili'u (after Queen Liliuokalani, who is said to have played one very well and who is considered one of the greatest Hawaiian songwriters, having written "Aloha O'e"  and many other great songs.)  

 The tenor uke (typically 26" long) is the size most often played by professional uke players. (Israel Kamakawiwo'ole's famous "Over the Rainbow" recording was played on a 4-string tenor with a low (wound) "G" string.) Lyle Ritz tunes his tenor to DGBE, the normal tuning for a baritone. Most tenors, though, are still tuned to the traditional  re-entrant  "My Dog Has Fleas" tuning, or GCEA with a high "G", same as soprano and concert sizes. Re-entrant tuning means the outside strings are only a whole tone apart, and higher than the middle two, so that either an up- or a down- stroke will hit a higher string first.

   The soprano, or "standard", ukulele is still the most popular size. This is partly because it's the least expensive and most portable. Tiny Tim played a soprano. You'll see some really LARGE players playing the soprano quite well, although guitarists and folks with fat fingers will probably be more comfortable with one of the larger sizes.

     Between the standard (soprano) and tenor sizes is the concert size. It has a fuller tone and more frets than the soprano (typically 15 as compared to 12 on the soprano and 19 on the tenor) while still being small, light and highly portable. For some reason, it has never been as popular as the other sizes, although it's gradually catching up in popularity. Concert ukes can be found in 4 and (occasionally) 6-string versions. All of today's fine ukulele builders make concerts, but vintage collectible concerts seem harder to find and therefore more valuable and expensive than other sizes.

   I think of a baritone ukulele  as a cross between a uke and a guitar, or a small nylon-strung guitar with only 4 strings. The first baritone uke was built in the mid-50's for Arthur Godfrey. The new size and sound seemed to catch on quite rapidly, and by the 60's most uke makers were selling almost as many baritones as sopranos!  The baritone is typically tuned a fourth lower than all the other ukuleles, like the 4 high strings on a guitar (EBGD), although there are string sets available which allow for standard (GCEA) or low-"G" ukulele tunings. Baritones can sometimes be found or special-ordered in 6 and 8-string versions. There is also an instrument called a guitarlele, which is a baritone uke with a wider neck and 2 bass strings added (E and A), making it a miniature nylon-string guitar.

     The tiple, (spanish for "little guitar", originally pronounced "tee-play"), is not technically considered  an ukulele. Its 10 metal strings are tuned GCEA, like the uke, but in 2 sets of two and 2 sets of three, so the chord positions and fingerings are the same as the uke. It's usually played with a pick, and has a sound somewhere between a mandolin and a capoed 12-string guitar.

Here's a link to Part One:

http://reviews.ebay.com/The-Ukulele-Hawaii-apos-s-gift-to-the-world_W0QQugidZ10000000000865181