I have verified that the thermostat is working, but cannot get the a/c to turn on. The outdoor unit appears to be OK, and the cooling lines into the house are cold, I can hear the electrical system "humming" but the blower will not turn on. I turned the system to heat and the blower DID kick on, but the system did not ignite. Any help would be appreciated"

The indoor blower is not working? Did you put it to fan "on" at the thermostat ? You may have a bad fan relay , fan motor or possibly a fan motor capacitor.

Fan "on" at the thermostat did not get it to kick in. But when I went to the heat cycle, it did...

The heat uses a different relay and motor winding than A/C and fan on. It could be the relay. That would be a simple fix, if it wasn't part of the control circuit board.

How do I determine if it is relay and/or part of the control circuit board? This is a Lennox unit (about 4-5 years old)

Look in the furnace , if it has a printed circuit board then it will not have a separate relay, put a wire jumper between the green and red thermostat wires in the furnace and the fan should work, if no fan then you would have to check with a volt meter to the motor windings to see if the board is asking the motor to run.