The baritone ukulele is the largest of the four main sizes of uke with a scale length (the distance between the nut and the bridge) of 19 inches.
As well as being larger than the other ukuleles, it is tuned differently. While other ukuleles are tuned gCEA (with the G being higher than the C), the baritone ukulele is usually tuned DGBE - exactly the same as the top four strings of a guitar. This leads some people to say that the baritone is closer to being a small guitar than a large ukulele.
Advantages of the Baritone Ukulele
Plenty of Bass: Compared to the other ukuleles, the baritone has a much larger bass range. This will give you many more options when accompanying yourself and more 'open' chords.
Bigger Sound: The larger body of the baritone gives it a louder and deeper tone than the other ukuleles.
Very Like A Guitar: Many guitar players who wish to switch to the ukulele choose the baritone first. With the strings being tuned the same as a guitar, there's no need to learn new chord and scale names.
Disadvantages of the Baritone Ukulele
No Ukulele Sound: The sound of a baritone is a lot closer to that of a guitar than a ukulele. You certainly don't get the close harmony chords you get on the re-entrant tuned ukuleles. And the sound of it when strummed isn't that same sharp, punchy tone you can get with the smaller ukuleles.