And to round things out there are also breadboards, where you can stick in components without soldering and there are modules, which are usually just one Chip with minimal components added and some pins so you can stich them easily into a breadboard.
Breadboard (from WikiPedia
Module (from Adafruit.com
If you start out, get a breadboard and some jumper wires. You cann connect the sockets on your Arduino to one of the holes on the bread board and tinker around to your heart's desire. Cricuits on Breadboards are a very temporary affair and don't hold up well to permanent use. All those jumper wires get in the way easily.
Prototype boards are already more permanent and useful if you have your circuit ready to be used in reality but it's not worth creating a circuit board.
Shields are designed for more permanent use and often offer all necessary functions for an application (eg Motor drivers, GPS). If you need that function and don't want to reinvent the wheel, they're very useful, but in my opinion they're too expensive to buy without an application in mind.
Modules are usually easy to use single function things to add one function. They can be added to breadboards and prototype boards just like other electronic components. Often modules are created to allow easy use of chips only available as SMD components.