Looks like your bond beam is cracked and half the tiles are fallen off. You should call your local renovation companies to get some quotes.

No, the bond beam is at the top of the tile shown here. The previous owner poured a concrete patio on top of it.

The other half of the pool has the original concrete edging for the pool with the same tile. None of it is missing. My thought is they were going to do something to dress up the edge, but I just don't know what.

If you could find that tile you could put a row on top of the other... double tile rows. That would dress if up and take away the appearance of concrete laying on top of your pool edge.