I weld mild steel at home to make things I want and to repair things I need.I do not weld aluminum or stainless steel. I make very little artwork with my welder. I am largely self-taught. I am aware of Tim Anderson's Instructable: Cheap Welding for Punks. He included some information I will not cover. I think I have some information he did not cover.
If terms in the next paragraph are new to you, see the Definitions paragraph that follows it. The italicized words are defined there.
Most of my experience is with electric stick welders. They cost less than other welding systems, but require more practice and skill to learn well. Many of the things one must do to get a good weld with a stick welder are not much different from a similar situation using a wire feed welder (penetration, weaving, preventing sag), and I will discuss both at the same time when possible. I will mention a little about oxy-acetylene (gas) welding. But, for the beginning, occasional welder; a wire feed welder makes learning to weld much easier. MIG is generally preferred and more expensive than flux core wire feed welding. Both have their place. People who want to begin welding are often short on cash, and may well choose to begin with a stick welder before possibly moving up to a wire feed welder after a few years, so some attention will be given to stick welding. I have no experience with TIG welding and will not discuss it.
Stick welders have this name because the coated wire welding electrode resembles a stick you might pick up from your lawn.
Wire feed welders use a continuous wire electrode on a spool. Rollers in the welder driven by a motor feed the wire at a steady rate through an electrode holder usually called a gun. Wire feed welders were invented to increase production rates by removing the need to stop and replace a burned electrode.
Penetration means the welder did not merely lay a bead of welding material over the top of a joint, but some of the parent metal melted and fused together below the weld, too.
Weaving is moving the arc from one side of the joint to the other in order to make certain the weld penetrates into both sides of the joint. A weaving pattern is also used to prevent sag when welding in an upward direction on a vertical joint.
Sag is molten metal that flows out of the joint while welding due to the effects of gravity. This can be a problem when the weld is vertical, overhead, or horizontally along the side of a vertical surface. Avoiding excess heat, electrodes designed to harden quickly, the angle at which the arc is directed, and general manipulation of the arc are used to control the weld and counteract sag.
Oxy-acetylene welders use two tanks, one filled with oxygen and the other with acetylene. Both gases flow through hoses at controlled pressures to mix and burn with a very hot pointed flame in a torch.
MIG refers to a wire feed welder that protects the fresh weld from mixing with oxygen in the air by continuously covering the weld area with an inert gas. If oxygen comes into contact with the hot, fresh weld; the weld becomes porous and is weakened.
Flux core welders use a thin wire electrode that has a chemical powder in its center. This powder melts and flows over the weld to protect the fresh weld from oxygen. This hardened coating is called slag. After the weld cools, chip or brush it off, depending on how thick it is.
TIG welders use a torch with a sharp tungsten tip to make a pool of molten metal with an arc. The operator dabs metal from a thin wire rod into the pool. At the same time the operator increases or decreases current to the arc with a foot pedal control. TIG welding makes beautiful welds on aluminum and various special metals. It requires much skill and practice to learn.
Other terms you may see in welding literature (Do not worry about these, unless you encounter them):
SMAW - shielded metal arc welding - stick welding
FCAW - flux cored arc welding
GMAW - gas metal arc welding - MIG welding
GTAW - tungsten inert gas welding - TIG welding
(The photo is from Bing Images.)