While there are many modes of operation in Satellite communications, my main focus currently is on the FM Repeater Satellites (Voice).  Currently two Satellites provide this mode:
UO-14, and AO-27.  Both of these are easily accessed using a dual band (VHF/UHF) HT, Mobile, or Base Station.

The trick is to know when the Satellites are going to be overhead (in view).  Once you have that, it's just a matter of tuning in the proper uplink and downlink frequencies, aim your antenna, and wait for a break in the action so that you can transmit your call sign and grid location.

I may have already lost a few of you, so let me explain the above a bit more.
As I said, the trick is to know when the Satellite is overhead (in view).  This is the point which the Satellite is the closest to you.  The easiest way to determine this, is to use your Home Computer, and install one of the many different Satellite tracking programs available to Amateurs (see Directory of Related Links).

Once you are equipped with a tracking program, and updated it's Keplerian Elements, you are now ready to program your Radio's channels to the appropriate frequencies. The setup for AO-27 would be the following:
Uplink (TX): 145.850 FM
Downlink (RX) 436.795 FM
Watch your tracking program, and when it tells you that the Satellite is above the horizon, begin monitoring the downlink frequency for activity.  The Doppler effect will cause a shift in the Satellites frequency as it progresses from horizon to horizon.  So to accommodate for this effect, program three RX channels into your Radio.
One for just above the horizon 436.805,
One for approaching you 436.800
One for overhead 436.795
You can add more as you begin to get the feel of it, and switch between them as needed for best reception.