Quarter Pipe and Half Pipe
Quarter pipes and half pipes are named after their shape. A quarter pipe looks like the inside of a quarter section of piping, and can take a bike rider from street level to vertical. A half pipe is like two quarter pipes stuck together, sometimes with a short flat section between. Quarter pipes are ideal for aerial tricks and can also be used for some rail tricks. Half pipes have the same uses as quarter pipes, but give the rider the advantage of going back and forth, doing tricks continuously.
Jumps
Jumps are designed to let BMX or mountain bike riders jump over an obstacle or a section of ground. They can be built with straight surfaces, in which case they are called wedges, or with a concave surface. If a jump is concave, it will generally give a higher jump at the expense of distance. Wedges provide longer distance, but don't give the same impressive aerials.
Fun Box
Fun boxes are actually combinations of jumps or wedges. Two or four ramps meet at a horizontal surface in the middle. Riders can jump over the box or ride to the top, then use the downhill to gather momentum. Many fun boxes also have rails extending from the top, which riders can use to do grinds and other tricks.