From your question we will ignore "removed material" forming of sword where a blank of metal is milled or ground down to a final shape and cast swords that were formed from molten metal in a mold and ground to a final shape.
A forged sword is created from a billet of metal(s) heated and mechanically shaped with a hammer until it is in an close approximation of the final shape. Final shaping polishing and sharpening is a separate operation.
The iron or steel used in the sword may be a single block of metal or built up from iron wands in the Norse tradition. Alternately the block may be folded and refolded to build up a laminar structure that enhances strength and flexibility in the Japanese tradition. Similar prcesses can be used in producing a Damascus blade.
Some swords are fabricated from various grades of steel each with a desirable attribute (strength, flexibility, ability to take an edge) located at various parts of the sword (tang, edge, core, cover, back). These are hot welded together to make the final product. This prevalent in the Japanese tradition.
Modern "society swords" and "replica swords" may be forged from an approximately right shaped billet in a drop forge