One-Piece vs. Two-Piece
One-piece softball bats use the same material throughout the entire design. The advantage of the one-piece design is that you get a stronger, stiffer bat that is generally favored by power hitters looking for as little flex as possible. In two-piece softball bats you'll find that the handle is a separate piece from the barrel and that the two are bonded together. The advantage to a two-piece bat is that it's design allows the barrel to flex at the point of contact creating a trampoline effect off the barrel. Two-piece bats generally have less vibration in the handle due to the separation of the handle and barrel.
Alloy vs. Composite vs. Hybrid
Alloy bats are generally constructed with a one-piece design out of aluminum or aluminum that is mixed with other metals to make a stronger product. The advantage to this strength is that it allows alloy bats to have thinner, more responsive barrel walls. Composite bats, on the other hand, are made out of a mixture of carbon fiber, graphite, fiberglass, and sometimes Kevlar. Hybrid bats feature a two-piece design in which an alloy barrel is bonded to a composite handle. This makes the handle lighter and allows the alloy barrel to be made longer than on a traditional alloy bat.
ASA vs. USSSA
There are very few fastpitch softball leagues that are governed by anything else other than ASA rules. Fastpitch leagues governed by USSSA, NSA, or ISA rules do exist though, and there is one main difference in bats made for these leagues: higher performance. For instance, ASA approved bats must adhere to the 98 mph batted-ball speed standard, while bats approved for USSSA and other associations are allowed to have a much hotter 100+ mph batted ball speed and a bat performance factor (BPF) of 1.20 or less. You can find a full rundown of rules and regulations here.