He he, to me, shooters are the traditional horizontal and vertical-scrolling (and even isometric ones like Zaxxon or Viewpoint) games, with a space ship or other vehicule that dodges bullet and fires back. There's no control over progress, the game just slowly leads you to each point and you just have to survive through it.
Games like Metal Slug, Gunstar Heroes and Contra feature characters, walking on foot, often similar to platformers, but more straightforward and with an emphasis on the shooting part rather going to A to B to C. Those games are sometimes called action games, but I prefer the term shoot-em-ups to differentiate them from real action games like Shinobi (platforming, shooting, close range attacks) or beat-em-ups like Final Fight (plain close combat versus multiple opponents).