In the old times, I used to play some games on my Amiga 500 where the player needs to continually shoot and advance. The ammo was unlimited. What is this genre called in gaming terminology? I can give Silkworm as an example.
This is called a "shoot 'em up," although there are several related sub-genres:
Shoot 'em ups encompass various types, or sub-genres. In a "fixed shooter" such as Space Invaders, the protagonist can only move across one axis and enemies attack from a single direction. In a "multi-directional shooter" the protagonist may rotate and move in any direction. By contrast, a "rail shooter" protagonist is viewed from behind and moves "into the screen", while the player retains control over dodging. "Tube shooters" feature similar viewpoints, and their protagonists fly through abstract tubes. "Scrolling shooters" encompass both "horizontal shooters" and "vertical shooters" (featuring side-on and top-down viewpoints respectively) and in turn "bullet hell" games and "cute 'em ups". "Run and gun" games feature protagonists on foot, rather than spacecraft, that often have the ability to jump; they may feature either scrolling or multidirectional movement.
By this definition, this game would be a horizontal scrolling shooter.
There are a million games in this genre, including classics like Galaga, and Gradius. Elements of rail shooters can be found in many games - if you've ever had a vehicle segment where you were the gunner and someone else was driving, for instance, although arguably one of the most famous rail shooters is Star Fox. Bionic Commando's top down segments would be a good example of a "run and gun" game.
Another term they are identified by is 'shmup' which is just a shorter form of shoot 'em up. http://www.shmups.com/ is a pretty good website dedicaded to (mostly) retro shmups so you may find it interesting.
PS: If you are interested in the genre you may check out the 'Project Touhou' saga :)
The genre of these kinds of games are "Scrolling Shooter"s. But to just tell you that doesn't tell the whole story. Hence, Wikipedia:
"Scrolling shooters" include vertical or horizontal scrolling games. In a vertically scrolling shoot 'em up (or "vertical scroller"), the action is viewed from above and scrolls up (or very occasionally down) the screen. This has the advantage of allowing complex patterns of enemies as well as allowing even simple graphics to function convincingly.
The other main type of scrolling shooter is a "horizontal shooter" or "side-scrolling shooter", in which the action is viewed side-on and scrolls horizontally.
A "Shoot them up" with horizontal scrolling.