Bought two 1 7/8" trailer balls from different stores that I installed on two different vehicles, two different manufacturers.  One is tight and the other is very loose when I hook my trailer up. Is this a common thing or a fluke?

Is the loose ball on the older vehicle?  Just a guess.  

No. On the newer.

What exactly is loose? The ball itself, there is play in the trailer when attached? or the bar the ball is attached too?

Just a guess, but look to see if there is a difference in the length of the shank (part between the bottom of the ball and the bottom flare).  If they are different, I'd bet the long shank ball is the one with the problem.

What exactly is loose? The ball itself, there is play in the trailer when attached? or the bar the ball is attached too?

Play in the trailer on one ball, tight on the other.

OK. I tightened up the nut on the trailer right where the ball fits and that took up the huge amount of "slop". How much play should I leave in it or should it be as tight as possible?

What is the proper way to lube the ball?? I can see that if you grease it up, it will attract dirt while towing and that will just grind away the surface of the parts that are making contact.

Use a grease gun to put grease inside the coupler on the trailer.  I'll hit it with 2-3 squirts every couple of months.

Do you think that shiny ball is going to stay pristine?  The ball on my truck now has been in use for a little more than three years.  Even with regular greasing, it looks pretty bad.