A regular taper I assume is what we call a conical or euro taper. It's the taper used on snooker cues & billiard cues. The taper angle is constant from joint to tip, making it very stiff & rigid while still allowing for a small diameter tip.
A progressive taper is just that, a taper angle that progressively changes from tip to joint. Most have a straight area from the tip back to a particular length & then begin to get progressively larger, progressively faster. The taper is actually curved. The idea is to provide a smooth stroking area for the player. As such, it has much more flex than a euro taper.
There are infinite combinations & variations of these two tapers. Every builder has a unique taper shape they prefer. For me, it's a compound euro taper. My taper has two distinct euro tapers joining at around 11" from the tip. It's just what I found to work well for me. Everybody has something specific to them.