The ukelele is a small member of the lute family of instruments (meaning that the resonating chamber does not span the full length of the instrument) similar to the guitar. It is a Hawaiian instrument, a derivative of the Portuguese machete de braga, that has spread to numerous other Pacific Islands and has gained popularity in variety of genres of modern folk music in the US and worldwide. 
The ukelele appears in several sizes, ranging from the most common, the soprano, to the rather hefty baritone, which is about half the size of an acoustic guitar. My Ukelele is a tenor size, which has an almost identical sound to the classic soprano, but with more volume (its size is compared to an acoustic guitar case in the image below).
There have been many prominent ukelele players, but one of the most iconic was Israel Kamakawiwo'ole (pretend the w's are v's, and then google how to pronounce it). His improvised recording, embedded below, of a medley of "Somewhere Over the Rainbow" and "What a Wonderful World" has a combination of rawness and purity that immediately burns it into the memory of anyone who hears it. 
The ukelele differs from the guitar, besides the obvious size disparity, in that it has only four strings to the guitar's six, and is used almost exclusively as a strummed, chord-playing instrument, while the guitar is used almost as much for melody as for chords.