TIG: "Tungsten Inert Gas" There is an electrode holder that has a tungsten rod in it. This rod is not consumed by the welding process. The arc jumps from this rod to the workpiece. Usually, there's a foot pedal, or a thumb-actuated current control to allow you to vary the current the arc is producing. The arc is surrounded by a flow of inert gas, either argon, helium, CO2, or a mixture, that prevents the metal melted by the arc from oxidizing. Once a puddle is flowing, fill rod is added by hand to maintain the weld bead. It's like oxyacetyline welding, but the heat to melt the metal is provided by an electric arc.
MIG: "Metal Inert Gas". The metal for the weld bead is in the form of a wire, that is melted by an electric arc as it touches the part to be welded. It's fed in a constant-speed advance through the center of the conductor that supplies the electric current. Voltage, and wire speed are controlled by knobs on the machine. Shielding of the arc is supplied either by a flux made into the wire (flux core), or by CO2 or Argon piped down the stinger lead to the welding tip. Plain wire is used on gas-shielded equipment, which makes a smoother bead than flux-core wire.
Both types of welders have their advantages, but you'll need some training before you're qualified to do good work with either one.
Jerry