Most of the information above is correct. Some clarification:
Casting: Molten metal is poured into a mold, and has grain structure that is more related to flow and cooling that final part use. It is cheap and easy to manufacture in large quantities, and in many applications works perfectly fine. There is typically a number of post-casting processing such as grinding, polishing, or machining. Cast metal can be prone to porosity, making them potentially brittle and/or weak. There are also different kinds of castings, e.g. sand (least expensive tooling), investment casting (more expensive tooling but better detail on final parts), and die casting (most expensive tooling), the latter typically used in soft metal applications.
Billet: Starts in ingot form (like al metals) as a big cast bar that is hot or cold rolled into plates or bars. The grain structure is aligned along the length of the bar, so aligning the grain to the final application before machining optimizes the strength of the part. This is the most expensive way to mfr parts as there is typically lots of waste. Porosity is not normally a problem with billet.
Forging: Forged parts begin with round bar stock heated to red hot soft, like very firm putty. The long bar is placed in a forging die (two part mold) and is hammered once or twice into the mold. The grain structure is nicely alignined (usually) with the final part, making it the strongest of the three mfg methods. Usually forged parts are then partially or fully machined (like a billet part), but at lower cost since the raw forging is close to final part shape.
There are other techniques that each method can use to further optimize function, strength, appearance, etc., but this gives the basic ideas of each.