[When I push the remote to open my garage door, it opens as far as about three inches, then stops. So, I push it again to bring it back down and start all over again, and it works right the second time. It's behaved this way several times this last week. Can someone tell me what's wrong?

Could be a flaky limit switch or photo sensor. There is one that signals the opener not to reverse when the door hits the floor.

It could also be something with the tension on the door. I think I remember my dad doing something like that when he replaced his, and then just had to adjust it. Maybe if there is a certain amount of resistance, it just stops moving, but then if you bring it back down again, the springs don't get as tight before going back up again? You could try to loosen the tension a little (if possible) Just a thought, definitely not an expert.

I looked at the troubleshooting page of garagedoor.org, and I found this:

3. Door closes part way and reverses.
Cause... Solution
Broken Cable... Replace cable.
Close force not properly adjusted... Adjust force.
Dry rollers, bearings... Lubricate.
Misaligned, dirty or defective photo eyes... Align, clean or replace if needed.
Track is bent.. . Straighten, adjust or replace track.
Worn Extension Spring Cable pulleys.. . Replace worn pulleys.

I hope this was helpful!

jjmus1 's answer sums it up pretty well. But I am leaning to an improper adjustment on the limit switch, or the switch is loose.