I used to take a few woodworking classes in which we used the cabinet table saw, a chop saw on a rail, and a router. I also got to use a jigsaw. However, I have seen a strange hand-held power tool that seems to act like the chop saw on rails and a jigsaw. It has a metal platform on the bottom, and the blade is on the left hand side of the handle. So the blade cuts down and back through the opening in the metallic platform. I believe you turn it on and then push the guard back with a lever or something, so as not to risk getting your finger cut off while the tool is on. They roughly make the same noise that a chop saw or miter saw, as they are sometimes called, make.
Is there a name for this particular cutting tool used by people needing to cut into walls to fix things inside?
From this part of the question, it sounds like you're talking about a reciprocating saw. This type of saw has a long body with a straight blade protruding from the front. The blade travels forward and backward, much like a handheld saw does. These saws easily cut through anything you're likely to find in a wall: drywall, wood, nails, screws, pipes, metal studs, etc. Milwaukee's versions of the tool are named Sawzall, but sometimes you'll hear that name used to refer to reciprocating saws from other brands too.
It has a metal platform on the bottom, and the blade is on the left hand side of the handle. So the blade cuts down and back through the opening in the metallic platform. I believe you turn it on and then push the guard back with a lever or something, so as not to risk getting your finger cut off while the tool is on. They roughly make the same noise that a chop saw or miter saw, as they are sometimes called, make.
This part of your question is almost certainly referring to a circular saw. Sometimes these are called Skil saws, again after a popular brand. Circular saws most often have the blade to the right of the handle, but there are also versions available with the blade left of the handle. The bottom of the saw is a metal plate called the "base plate" or sometimes the "shoe", and a motor is mounted to this plate. The circular blade fastens directly to the output shaft of the motor and protrudes down through the base plate. The motor can tilt to the side on the base plate to make angled cuts, and it can pivot forward to reduce the depth of cut. Front and rear handles are attached to the motor housing, with a trigger switch built into the rear handle. A spring loaded blade guard covers most of the blade when not cutting, and sometimes has to be pulled back by the operator to start a cut.
A variation on the circular saw is the worm drive circular saw. It's the same idea, of course, but the motor sits somewhat behind the base plate, with the rear handle attached to the back of the motor. The name comes from the fact that the motor turns a spiral "worm" gear that drives the shaft that spins the blade.
Circular saws of either variety are used heavily in construction to cut lumber, and they can be used to cut holes in the exterior of a building to add a window or something. But if you were going to cut into a wall to fix something, or even to remove the wall entirely, you'd probably go with a reciprocating saw.