Martin are probably the number one name to look for In Ukuleles, they are a byword for quality and, certainly for older models, command the highest price, (the new ones aren't cheap either). 

Martin first made Ukuleles in 1907, (I have read they were made for Bergstrom the big Honolulu Music shop) but the early versions were apparently not very good because they made them too much like a Guitar, however in 1916 they had another go, got it right and became probably the most influential Ukulele maker ever. They also produced different levels of decoration for Ukuleles starting the Style 1 and 2 Sopranos in 1916. S2's had triple bound top binding (black-white-black) with ivoroid outer binding and S1's had no binding around body and a small ring around soundhole. In 1918 they added an outer binding to the S1 and the unbound model became the S0, they also started the S3 with a 7 layer top binding, alternating stripes of white celluloid and ebonized maple and a 5 layer soundhole ring. In 1920 they started making them of koa as well as mahogany and these koa variants were knows as the S1k S2k and S3k, (they never made a koa S0), and in 1922 they started and even fancier koa model called the S5K, (there never was a style 4? and there was only a mahogany S5 made in 1941 and 42 after production of the 5K finished in 1940). 

Martin inadvertently created two of the larger scale definitions we use today. They started out making Taropatchs in 1918, of which people liked the extra volume but didn't like the tuning issues so in 1925 Martin put the strings back to 4 and the Concert was born. Other tried to call this scale length a Tenor at the time but Martin stuck with Concert and a couple of years later in 1928 they started making 17in scale Tenors as a 4 string version of their Tiple Ukuleles, (They didn't come up with the Baritone scale though), In addition to the scale lengths , in 1919 they created the 10 string Tiple Ukulele, (initially for the music publisher/distributor William J Smith & Co but mainly sold under their own name). Concerts, Tenors and Taropatches were only mass produced with S1 decoration in mahogany, though more decorated and koa custom models were made. Tiples had a different method of designating the level of decoration to other Ukuleles and I can't say I really understand it? All the levels start with a T but the first one is the T-15, then a T-17, T-18 and T-28 and I think it was just a wood change that made for a different number. For example the difference between the T-15 and the T-17 is the T-17 has a rosewood bridge and the T-15 ebony. Another thing to note about this style designation is it wasn't adopted until 1923 so anything from 1919 to 1923 was just a Tiple (apart from, as a precursor to this T numbering in 1922 they made a Koa Tiple they called the T-45 and one called a T-42? they also made a few 8 string Tiple/Taropatches? in 1922 as well?).

Taropatch production finished in 1931, Tiple production finished in 1936 and all of the koa models finished in 1940 but some production of the mahogany models continued through WWII albeit going back to using wooden friction pegs, (due to lack of metal), that they had stopped using in 1923 on top models and 1926 on the rest. After WWII mahogany production increased again with more S0's being produced in 1950 that at any time since 1926. Also Tiple production was restarted in 1947. In 1960 the Baritone S1 was introduced, but generally sales were falling off and in 1965 production of T-15 and 17 Tiples, Concerts and S1 and S2 Sopranos was stopped. The rest continued in production until 1977 but after that Martin only produced Ukuleles as a custom item.

In 1997 Martin started to produce Ukuleles again but made in Mexico and only the Backpacker Soprano as part of the Backpacker range of instruments. In 2006 Martin stopped Backpacker production and decided to go back into production with more usual Ukuleles, still making them in their Mexican factory rather than in Nazareth, Pennsylvania, and since 2010 they have a fairly eclectic mix of some of the old favourites plus some new wood options including cherry, and a laminate they started using for their X range of Guitars. One thing though they seem to have gone back to having an S1 with no outer binding and no S0. Whist they have restarted making Sopranos Concerts and Tenors there are no Baritones, Taropatches or Tiples, apart from custom options.

Unlike their Guitars and Mandolins, Martin never put serial numbers in their Ukuleles, (apart from a few of the very early ones), so they can’t be dated as accurately as other Martin instruments, however there are lots of sites on the Internet that give a fairly full run down both of the history of Martin Ukuleles and all the tips of how to date them so I won’t go into it all here apart from a couple of obvious things. Until 1932 there was no logo on the front of the headstock just a stamp on the back, and after 1934 they took the stamp off the back. In 1962 Made in USA was added to the sound hole stamp and the latest logo is embossed father than just a straight decal.

Martin did make some Ukuleles for others to brand, like Wm J Smith & Co.,(who also used the name U-Ka-Lu-A on their sound-hole labels; even in Martins!), Grinnell Bros. (with their Wolverine branding), Wurlitzer and Ditson, but not as many as people think, so there are often stories that brand "x" is really a re-branded Martin which aren't true. Again if you look on the Internet you can find out exactly how many Ukuleles Martin made in any one year both to be branded as Martin or for others to brand, (they kept very exact records). It is the case however that Martin allowed its workers to come in and make instruments for themselves on their days off. I don't know the full deal here but I have seen a few, very Martin like, un, or only partially branded instruments including Ukuleles that were produced this way. Mike Longworth is probably the most famous of these staff makers, (though he did continue making Ukuleles after he retired from Martin), so there are some “almost Martins” outside of the official records and sometimes using different woods to mahogany or koa.

Goya was originally a Swedish brand of Guitar made by Levin. Martin brought the company in the early 70's, shut down European manufacturing but kept the name going to brand instruments, including Ukuleles, made in Korea and Taiwan from 1980 until 1996 (though actual production may have finished by 1993?) They also had a branding, Sigma that they used for Far Eastern produced licenced instruments, (they later sold this branding on to AMI), but I don't think Martin ever produced Ukuleles under this name.

To my knowledge Martin never made any Banjoleles, and though at one stage they owned the Vega name, no Banjoleles were made for them at this time.

C.F. Martin were also the original "Lawsuit Guitar" victors because prior to Gibson's threats of lawsuits Martin DID take K. Yasuma & Co., (a Japanese chordophone maker), to court in the U.S for "copyright infringement" and won! The victory and subsequent penalties for Yasuma both emboldened Gibson and made non U.S., (especially Japanese), instrument makes very afraid.