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The roller coaster is a specialized rail system consisting of a track that rises and falls at varying rates. Sometimes roller coasters have inversions (such as vertical loops) that briefly turn the rider upside down, depending on the intensity desired for the ride. The track is not always a complete circuit, rides like shuttle roller coasters demonstrate this. Most roller coasters have multiple cars in which passengers sit and are restrained. Two or more cars hooked together are called a train. In today's high intensity, high adrenaline society people are looking for the next near death feeling. Below are some images of roller coasters found in the US.
As one can imagine there are a lot of calculations that go into making a roller coaster not only safe, but exciting. some key things engineers need to know about are momentum, potential energy, kinetic energy, work, and force.

This diagram shows one of the most essential forces that are required during an adventurous ride. The centripetal force is what keeps the railcar pushing out against the track. along with that the engineers need to make sure that the railcar has enough momentum to complete the loop without stopping at the top and falling to the ground below.
http://www.universetoday.com/wp-content/uploads/2009/11/centripetal-force.jpg Centripetal Force

http://www.thecoastercritic.com/wp-content/uploads/2010/04/Intimidator-at-Carowinds-Trip-Out.jpg
We all know it takes a lot of energy to walk up a hill. Well at amusement parks it COSTS a lot of money to push a railcar up a hill so instead they try to harness all the energy they can from it falling in the form of Potential energy (energy given by gravity and height above ground) transfering into Kinetic energy (energy by a mass moving).

FUN FACT:
Now most people run to the front of the ride when the gates open but is that where the best place to ride is? Well according to a study by first year physics students from the University of Gothenburg, Sweden; the best place to sit is in the middle for fullest 0 gravity effect and greatest affect of a loop.