There are many doorknob hardware designs.  The question asked most often is how to remove interior doorknobs with hidden screws.  It's more obvious how to remove the styles with exposed screws.   This explanation covers only one type with hidden screws, but the approach is similar for many others.

The last Figure, at the bottom of the page, shows the complete assembly.  You may want to take a look at it before starting.

The first step is to  remove the knob on one side of the door.

The knob is held on by a spring loaded thin detent (button) just behind the knob.  The knob on one side or the other has the detent.  You press on this detent and pull the knob off with the detent depressed.  Figure 1 shows a narrow blade screwdriver being used to depress the detent to release the knob.  

In this case the detent was located on the bottom, but it can be located anywhere around the barrel of the knob.  

Figure 2 shows the keeper slot (hole) the detent snaps into to hold the knob in place. 

Next, remove the trim ring/cover.  This is the step
most people ask about.   Figure 3 shows using a wider, thin-blade screwdriver to pry off the trim ring.  

In Figure 4 you see a small notch in the trim ring.  

This notch should usually be located at the bottom as shown by the screwdriver tip in Figure 3.  Depending on who put the trim ring on last it may be in almost any position.  Some trim rings don't have a notch.  In these cases you should still try to slip/work the screwdriver (or a table knife) under the ring at the bottom as shown.  You may mar/scar the door with the tool while prying the cover off -- by working at the very bottom of the cover the scar is less likely to be noticeable/visible.

From this point on, the procedure is pretty straightforward.  Remove the clamping screws, the actuating shaft assembly and the far knob, then follow the steps in the following figures.

If the steps still aren't clear let me know and I'll try to improve this page.

Pull Out Latch Assembly