What does tapered or untapered mean?

Tapering a staff means that the center of the staff is thickest and the towards the ends they get thinner. By removing mass towards the ends and not in the middle, this makes the staff only slightly less heavy, but makes it MUCH easier and faster to rotate. For more of the physics end of this please read my blog post  What is the best length for my bo staff? Many karate styles prefer this, because it's a happy balance between weight and maneuverability. And to get a thick staff to work for most uses, tapering makes this an option. 

Untapered simply means that the width of the staff or thickness remains constant throughout its entire length. 

A 1,1/4" staff tapered means that at the center it is 1,1/4" thick, but at the endges it is around 1" usually. Sometimes a bit more or less. 

for a bit more information on the way the staffs are tapered, I taper approximately in thirds. If you split the staff in thirds, (which is the way your hands should grip the staff) where your hands touch the staff is approximately where the taper begins. I do not allow an abrupt section where it feels straight, and then you feel an edge and then you feel it get thinner toward the end. I don't personally like this because to feel the edge of the tapered section beginning on the staff to me is distracting when your hands slip up and down the staff. I do hide this by gently rounding the staff at that point near the third's sections on the staff so you will not feel an abrupt edge. 

Also, I don't do a pyramid taper. Which is a taper from the center where it is thickest, and then from the center it gradually and evenly decreases towards the edges, like two cones connected in the center. I like the center section to retain its thickness because often the section of the staff that blocks is between the hands, and I want the greatest strength at that section of the staff.

Another reason I start tapering around the thirds of the staff, is because I like the fact that your hands develop a 'feel' for where the taper begins. So, as you work with your tapered staff, your hands will naturally develop a 'memory' of where they should be based on the thickness. This is really beneficial to developing a feel for your weapon. And makes for quick and easy adjustments, based not only on your feel of the weight distribution, (as if your staff were slightly off to one side) but also from the feel in the palm of your hands which I find very beneficial. And this will happen naturally without training. It's a memory and your hands will gravitate toward holding the staff in the correct position because if one hand feels that it is holding a thicker section (too close to the center) and the other is too thin, even subconsciously they will maneuver to find that even thickness which means your hands are centered.