Apparently the blower motor, i.e. the fan that blows the hot or cold air through all the vent ducting and into the house, had stopped and was either too dirty and gunked up to get started again, or has a bearing going out, or something. However, once he gave it a push to get it moving, it stayed moving. So the trick for now is to set the fan from Auto to On and leave it there. The compressor, the part that uses all the electricity, will still shut down when the house reaches the desired temperature, so leaving the fan on all the time won't kill my electric bill.
This is a temporary fix, but an indefinite one. Could last a long time. However Matt also gave me enough information to be able to take this motor out of the unit and take it somewhere to get it rebuilt or replaced if I need to, and even this shouldn't be horribly expensive. So basically this is the cheapest and best possible outcome to the A/C breaking. So thank goodness for small things. :-)