The original Arduino is opensource hardware based on the Atmel Mega 2560 (?) chip. Buying an original board from Arduino runs $50 - $60 but supports the developer.
Everything else is primarily cheap clones, most of which sadly call themselves Arduino's and even use the Made an Italy silk screening, which is slightly dishonest depending on your personal belief's on the matter. Other companies have started to rebrand their boards to distinguish them from the official Arduino boards (i.e. Freaduino).
Any Mega2560 board should work fine as the brains of your i3, though the manufacturing quality (solder and components used on the board) is questionable and should be researched if you are not using an original Arduino.
Everything else is primarily cheap clones, most of which sadly call themselves Arduino's and even use the Made an Italy silk screening, which is slightly dishonest depending on your personal belief's on the matter. Other companies have started to rebrand their boards to distinguish them from the official Arduino boards (i.e. Freaduino).
Any Mega2560 board should work fine as the brains of your i3, though the manufacturing quality (solder and components used on the board) is questionable and should be researched if you are not using an original Arduino.