The ball valve for the sprinkler system cracked over the winter. I managed to remove the old one that was broken but can't figure out how to install the brand new one. When I turn it clockwise it screws on the left but unscrews the right part - and vice versa. I'm not sure what I'm missing here. Any advice would help. Thanks,
That's just the way screw threads on pipes work, clockwise tightens and counterclockwise loosens.
Since you don't have experience sweating pipes, another alternative is to tighten the valve to the set of threads on the right and cut the vertical pipe on the left below the elbow. Go to your plumbing or home store and get the necessary Sharkbite connectors and pipe to reassemble what you removed. You can unhook Sharkbite connectors so a union would not be necessary.
Personally, Learning to sweat pipes would be in your best interest.
The reason you don't have a union is that the previous installer assumed you would simply unsolder the elbow when you needed to replace the threaded valve. That's pretty much what they had to have done (well, really just solder the elbow after the threads were tight, the first time it was assembled.)
So, you can learn to solder and try to squeeze a union in there, or you can learn to solder and just disconnect the elbow, tighten the threads, and reconnect the elbow.
Either way works.