What is the name of the following chess opening? What are its pros and cons?
Moving your bishop like that is called a fianchetto.  There are a lot of openings which fianchetto the king's bishop; what you are describing sounds somewhat like the King's Indian for white.

The advantage of fianchetto-ing your bishop is that it very quickly puts the bishop on the long diagonal, its most powerful position.  However, it takes two moves, and weakens your kingside (specifically the f3 and h3 squares), giving your opponent potential areas to attack.  For example, a common plan for black in many openings (if his pawn structure allows it) is to force the trade of bishops by Bd7, Qc8, and Bh3.  Once the bishops are traded, your king would be very vulnerable to attack.

Traditionally (beginning with Steinitz), it was taught that one of the goals of the opening phase was to control the center by occupying it with pawns.  However, Nimzovich introduced hypermodernism, which states that the center should be controlled from a distance by pieces, not occupied by pawns.  Neither view is more correct than the other - even today, both have very strong Grandmaster supporters, and accepted opening theory draws from both sides.  I mention this because yours is a hypermodern opening.