ASW (spelled out when spoken, ae-ess-doubleyou) is an abbrevation for "Anti-Submarine Warfare." This refers to all activities intended to combat submerged or submersible targets at sea, as opposed to those targets on the surface (ASuW) or in the air (AAW). Typically used in the Navy (U.S. Navy, at least) community to refer to fighting submarines, or to describe the ability to do so. Units are described as 'having ASW capability' if they can engage such targets.
Modern U.S. Navy warships are multi-role, and therefore there aren't really any dedicated ASW warships. The preferred platform for undertaking ASW is, in fact, a submarine - since surface units suffer inherent disadvantages when facing submarines (they are easier to detect when moving, they can't change depth to hide behind thermoclines or terrain, etc.) A famous aphorism in the U.S. Navy reads "There are only two kinds of ships in the world - submarines, and targets." Guess which branch likes to say that.
Weapon systems or ship systems, also, can be described as 'ASW systems.' Ship-mounted torpedoes are generally considered ASW systems. Sonar systems are used to fight submarines, not other surface ships (usually) and are generally ASW. ASROC, depth charges, and more complex systems like LAMPS are all intended to fight submarines, and are hence ASW systems. There are defensive ASW systems like PRAIRIE/MASKER, and strategic ASW systems like SOSUS.
With the proliferation of highly-capable nuclear and conventional attack submarines along with the increases in capability of the latter brought about by AIP systems, and the U.S. Navy's continued reliance on large visible surface units such as CVBGs, ASW is a 'core competency' the U.S. Navy cannot allow to falter. How successful they are at maintaining this skill set is a topic of frequent debate, without actual combat to provide an unarguable 'score.'