The car had been running fine. I pulled into a parking lot to turn around and when I put the car in reverse the engine revved but it did not move. I put in in drive and the same thing happened. I have been in cars where the automatic transmission had failed before, but usually there are warnings like the engine turning over too much for the speed I am going, or just locking up. I have never had one that suddenly had the engine apparently disengage itself from the transmission, except when it suddenly lost transmission fluid due to a hose disconnect. What is likely wrong with the car? It does have transmission fluid.
It could be a number of things. The most obvious based on your description is a problem with the shift linkage. It could have come loose or misadjusted. Other things include the torque converter broke or an internal problem. If your car is rear wheel drive the problem could be in the driveline, broken u-joint or rearend. If it’s front wheel drive a drive axle could have broke or you have a stripped drive hub. With the limited info you about the problem and vehicle, all anyone can do is make a wild guess at the problem.
Just for kicks, what happens when you put the shifter into the lower gears?
With one person in the car (with foot ready on the brake pedal) can the car be pushed while in gear?
By saying it “does have transmission fluid” are you telling us it’s in the full range when warm, on a level surface, while idling in neutral? Does the fluid on the stick look bright red, or brownish red, and does it smell burnt? How many miles on the vehicle, and what is your ride? Details, brother. 
Well, it is a sort of level surface, very mild uphill slant, since the car won’t move, the fluid is brownish red and neither smells nor tastes burnt.
Also, yes in the full range in neutral. 127 K on the car.
Does it ‘feel’ like it’s shifting when you move the stick through the gears? As in, does the shifter feel like it always did? Can you get it to roll when in gear?
127K is young, but harsh usage (prior owner) could be a factor. You could try a fluid/filter change, but my gut says that isn’t it. What year/make/model?
Tangentally, my 1999 Jeep Cherokee would not shift into 4th (while it was still under warranty). Turned out to be a failed sensor or computer module.
You TASTE the fluid?
My mom’s Minivan was stuck in low gear no matter what speed she was going. I freed up the shifts by pouring in a few capfuls of acetone to cut the gum and varnish locking up the works.
I think the more important question is what did the OP expect old automatic transmission fluid to taste like?
Very common at the auto auctions where I work to have “no reverse”.
Is probably why the car is at the auction and not being sold off the lot. Hopefully some poor sucker used car dealer will not ask if the car goes in reverse, ----silnce it goes forward fine. --------and forward is the only way to go at an auction.
Mechanical reason I don’t know.
But it is super common.