I recently replaced a failing power supply in my old Dell PC and it was a bit challenging but rewarding too.  Ensure you have the right kind before you begin but basically it is going to be a simple process.

After you shut down the PC, open her up and blow out any excess dust (compressed air is better than your own hot air since you tend to blow moisture).

find your old power supply and look at it to ensure the case will fit your new one.  Before unplugging anything check to see if your new power supply has the same connectors (mine had two connectors for the motherboard, two for CD readers two for hard drives and one for a floppy

Once you know what is plugging into where, begin removing the old power supply (screws on the back of the tower or side.  You may have a latch or clip to release the power supply.  Once the old girl is out, place the new on in and check the alignment.  I had to modify my back case to allow the plug to pass through since I could not get an oem Dell power supply.  Re-mount the screws (1/2 way tighten).  Now as you unplug one cable on the old power supply, plug in the new one to ensure you do not miss anything.  If you mother board plug in is too short (20 pin) and your power supply is 24 have no fear you probably just need to slide off (not break off, just slide) the four pin connector to plug into another spot.  Or you may have an older board like me that only needs the ten pin (I think the extra four pins are for super video cards).

Once you have everything connected double check the connectors are firmly seated and close it up.  Now for the momment of truth.  Before firing it up

MAKE SURE YOU HAVE THE RIGHT VOLTAGE SELECTED

On the back is a little toggle switch for 115 or 220V.  Switch it to 115 (110v) or 220 and then reattach all cables and power it on.  If you hear any beeps (that you did not have before you replaced it) you may have an issue.  Mine powered up without any and now my PC is more reliable (less crashes). 

For a few pictures use this guide as well

downloads.emachines.com/service_docs/grey_powersupply.pdf

you PC may be different but it gives you a general idea.