When I was a child, I saw this game on an Arcade Machine in Freiburg (Germany), and I was very surprised.
I had already seen Shoot-them-Ups like Space Invaders and Galaxians before.
But the innovation of the game Gorf is, that it combines different shoot-them-up games in an adventure-like way.
Additionally, you have the chance to gain rank titles by playing through all five levels and restarting them at a higher difficulty.

The levels are:
Level 1: Space battles. A formation of enemies in the Space Invaders style.
Level 2: Laser attack. Two enemies attack you by lasers, the others are simply trying to collide.
Level 3: Galaxians. You probably know the original game. The enemies look and move in the same way. They are shooting more, but are nevertheless easier to evade, because the player's shooter can also be moved up and down.
Level 4: Space warp. A space warp generates some enemies, which are circling out and finally disappear outside the screen. Theoretically, you can finish this round without shooting, but who would want to miss the score from hitting them?
Level 5: Flag ship. The end boss, a well-protected fighting ship.
Level 6: Space battles. If you succeed to defeat the flag ship, you can restart from the beginning.
Level 7: Laser attack. ......

I remember, that the original Arcade Game had a voice commenting the fight, which is missing here.
But everything else is perfectly in the same way as the original Arcade Game Gorf.
By the way, the name "Gorf" is meant to be "frog" backwards.