They come in a variety of shapes, sizes, adjustable, break-away, fixed, material composition, and probably some options I don't even know about. 90% of them will be adjustable, with only a few brands that offer cut to length. Adjustable, meaning the T handle releases the shaft allowing you to pick that length you want. Length depends on what style of riding you are doing and you can play around with what works best for you. Paddles blades do come in different shapes, which matches what style of water you are running. Some will be large wide blades allowing more contact with the water for better power transfer in your stroke. Longer more narrow blades are common on river paddles, since you don't need as much power transfer and use it more like a rudder (steering). Most paddles are made of carbon fiber, fiberglass, and/or aluminum. There are a few wooden paddle out there too. Travel paddles break in half (or more) allowing it to pack into the board bag, making traveling much easier.
Photo below shows ya a few different paddle option out there. Travel, break-away, river, fixed, and a special double blade paddle (left to right)