Any Chess game with the opening move 1.e4 is called a King's Pawn Game (or Opening).
|Double King's Pawn Game (Open-Game).|
If black responds with 1...e5 it is called a Double King's Pawn Game or an Open Game, if black plays a move other than 1...e5 it's called an Asymmetrical King's Pawn Game or a Semi-Open Game.
The reason the King's Pawn Game is so popular is probably due to the fact that it allows for easy development of white's queen and bishop. A pawn on the e4 square can be highly useful because it occupies one of the 4 center squares and attacks the center square d5.
The most popular response to 1.e4 is 1...c5 (The Sicilian Defence). The Sicilian Defence allows black to fight for the center by preparing to meet a d2-d4 advance with ...c5xd4. The Sicilian is among the sharpest and most analyzed openings in chess. I will cover this defense soon.
Pros: Occupies the center of the board, it frees up a lane for the kingside bishop and allows for
Cons: The e-pawn has no immediate back-row guard.