Look for a set screw on the knob itself, if so, then remove the knob and the cover plate on one side will come off with some work, the screws are underneath that.

Could be you have to push in on that nail head thingy to release the knob.

I usually ruin the first one I take apart.

Depress that nail head looking thing while pulling on the knob and the knob will come off. Then you can pop off the escutcheon trim plate and the screws will be exposed. Some of the escutcheon trim plates are held on by friction and some have a little exposed wire spring that you have to either push or pull to get it off.

Also, some escutcheon plates are threaded directly onto the lock body.

You can also look around the rim of the escutcheon plate for a small (often about 1/8 inch? gap in the edge against the door.
You push in the small piece of metal in the gap and the plate (and knob sometimes) come off the shaft.

Thank you guys so much for your replies. I was so excited when I pulled the knob out, by pushing the nail head and thinking that I could take everything apart then. Nevertheless, after pulling the knob out, leaving the horizontal shaft, I could not find anything to further taking it apart. The shaft inside the knob had no screws or holes. I did find a very little gap around the rim of the escutcheon plate (first I did not know what an escutcheon plate is until I checked my home repair book. You all should be very proud of me on this. At least I understand what the term is :-) ). I tried inserting the tip of a flat screw driver into the gap, but it did not do anything. I unscrewed the face plate, but the entire face plate was connected to the latch bolt assembly. I could only pull it out1/4." Please help! What to do next? I am so frustrated :-(

You may have the type that randy427 mentioned. That type requires a somewhat special spanner wrench to unscrew the escutcheon from the knob spindle. Does it have a brand name stamped into it anywhere?

Ccecilm, no, there is no brand name stamped into it. Will I be able to find the special spanner wrench anywhere. By the what does it look like? I may have something similar or I might be able to create one

a few options here, look to see if there is threads on the shaft, sometimes the collar screws off, also, try putting the flat head screwdriver into the slot and pry up, trying not to damage the door.the reason you couldn't pull the face plate out is because it passes through the the door knob shaft, once you have the knobs out, you'll be able to take that out. take note of what you are doing, putting the new lock in will be easier, good luck.

Here are some links for installation types.

http://professional.schlage.com/pdfs/pc/Standard_Duty/A-Series/A-Series_install.pdf

http://professional.schlage.com/pdfs/pc/Heavy_Duty/D-Series/D-Series_Knobs_Svc_Man.pdf
>>> on this one goto page 38, illustration 7 to see the wrench

Thank you guys. No other locks would be as troublesome as the one I am having right now. I have taken out and installed locks before, but it was not like this time. I went to Lowes, and 3 people tried to help me, but they have never heard or seen this kind of lock before. Well, I am going to try my best and follow what you all suggest. If it's not successful, then I will just change doors. LOL........

You can probably apply enough torque to turn the escutchion by using large channellock type pliers.

THANK YOU GUYS ALL VERY MUCH FOR EVERY INPUT. YOU GUYS ARE WONDERFUL AND GREAT. I CANNOT SAY THANK YOU ENOUGH!!!!!

Finally, my locks were out!!!!!

When I was using a flat screw driver to pry it out, the leverage was not enough to push it open.

I tried every tool at home that might be useful to pry it. I finally tried the "paint can opener" (the paint store usually gives you one when you buy paints). It was a liitle to big for the gap around the rim, but the leverage was just right to pry the escutchion plate out. Then, I saw the two screws there on the inner plate. I was so happy that finally this !@#$%^&*() lock could be taken apart. Thank you all. I love you all!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Sounds like a commercial lock set, or a very high end residential.

Commercial lock sets rarely have exposed screws.
They use a variety of 'tricks' to hold the covers in place, and the handles also.
Spring loaded 'catches' are very common.
There will be nothing but a small hole in the handle to use a punch to release the catch.
Escutcheon plates will have a tiny gap (often about 1/8 inch long) with a small metal catch 'filling' the hole.
You push in the catch and the plate will unscrew.
A few have the catch up under the knob and release like a knob.
Some have been made that use a spanner to unscrew the escutcheon.
Sometimes the knob itself holds the plate on, but the plate will have metal on the back that mates with the lock body and prevents rotation. Some just have a tab that catches in the spindle for anti-rotation.

None of them usually require a great deal of force, so look carefully before breaking out a pry bar.

The commercial locks you described sound like what I had. I was so frustrated working on it. I went to Lowes and tried to find something similar on the display, so I could show the staff what i got and find a way unscrew them. There was no lock like it on the display. At least now, after I posted my problem here, I learned about locks more. I will not use a great force next time, if I have to pry some locks out again. Thanks