Ukuleles are very small with four strings. The classic ukulele style looks like a very small guitar. I have two of them and they're easy to play and lots of fun. They'll also easily fit in a pannier or backpack with lots of room to spare...
Banjos are much larger instruments with four or five strings and a drum style soundbox, similar in length to a guitar. I have a five-string banjo in my loft that my gran gave to me but I can't really play it.
The Ukulele Banjo is a ukulele that has been built along banjo lines rather than guitar lines so that it has a banjo drum-style soundbox.
There's also a very rare style of ukulele made out of the shell of an armadillo, but I forget what they're called!
Classic ukuleles are very cheap, starting at around £14, whereas banjo ukuleles are a lot more expensive.
A proper uke looks like a miniature guitar, and sounds a bit like one. It's rather quiet. It's nearly always played by strumming all four strings together.
These guys are experts at it and I reccommend you catch one of their gigs if you possibly can.
There several types of proper banjo, but they all have a drum-type body and a neck much longer than a uke has. There are lots of styles.
The tenor banjo is what you'll hear played by the likes of The Dubliners - a low pitch, played with a pick with a rolling style rather as one plays a mandolin.
The five-string banjo is the one you hear played with picks on three fingers at lightning speed. Think "Duelling Banjos" or the soundtrack to my video here. The secret to the speed of this style is the fifth string. It's got a little tuning peg all of its own halfway up the neck (well, at the fifth fret, actually), and you pluck it with your thumb.
Now, with most fingerpicking styles on whatever instrument, your thumb hits a bass note and then gets out of the way while your other fingers do the clever stuff over the top. This means that your thumb is idle most of the time. Having a high-pitched string on the bass side of the neck (that's the side closest to the player's face) means that, after the thumb has plucked the bass note, there's a high-pitched string it can pluck from time to time while it would otherwise be idle. This keeps the thumb busy and produces a much more comlplex, flowing sound than there would otherswise be.
A banjo/uke looks like, and is, a cross between the two: a short, four-string uke neck on a banjo body. You get the volume of the banjo (and, particularly with a resonator, a banjo is LOUD), with the other characteristics of a uke.