Arduino is Open Source, which means there are many different companies making Arduino and Arduino compatible boards. These often have the same form factor, but many do not. However, they mostly work the same way and can be programmed with with Arduino IDE (the program on the computer). Below are some that I happened to have laying around.
This is just a small sampling of different types of Arduinos available. They are simply the ones I had laying around. When using a new type of board it is important to orient yourself. Where are the digital and analog pins? Are there extra pins? Is there a reset button? Most types of Arduino clones will have all the standard pins. Some will have extra. These may be more input/output pins or they may be more pins to make serial or i2c communication easier.
It will also be necessary to determine if you need to download anything extra to make them work. The Teensy 3.0 picture above at the lower right requires you to download the Teensy Loader in order to work.