An Angelgotchi is a Tamagotchi created by Bandai back in the late 90's. It was first released in Japan.
It is one of the 'themed' Tamagotchis. This theme was Angels. The characters would be like angels, the game as a 'angel-themed' game, and many other angel themed things. It also came with a few extra fun things, such as fighting bats away, and Angelgotchi on stroll.
You can read more about them here: http://www.mimitchi.com/html/tangel.htm
On a side note, the Angelgotchi is the only Tamagotchi ever to have twins, and it is a character. There is no 'special cheat' to get it.