Sword forms are named forms - or "styles" - in which swordsmen are trained to use the sword. Blademasters and Gaidin have a wider knowledge and mastery of the forms. Like real-world parallels, they are not always meant to be used exactly as choreographed in the middle of combat and instead convey a named concept or tactic for the swordsman.
While training, one may use a practice sword instead of a real sword to move through these forms. A practice sword may be sword-shaped piece of wood, or a bundle of thin, loosely bound staves or lathes in place of a blade.
Some, if not all, sword forms are taught in the Seanchan Empire. When Rand al'Thor battles Turak Aladon, Rand recognizes the forms that the High Lord used. They were different from what Rand was taught, but similar enough. It is possible the names might be different as well.
List of known forms