This is the Zercher squat of the upper body – it's a great core strengthener on top of its main purpose.
The Z-press is when you do an overhead press while seated on the floor with extended legs. The purpose of sitting like this with no back support is to make the core work a lot harder to stabilize the upper body.
It also teaches you proper overhead pressing technique. If you push too far forward or lean back (the two cardinal sins of overhead pressing), you simply won't be able to use a lot of weight and your core will tell you that you're off-line.
You can get a lot out of the single-arm version. Most people get a better mind-muscle connection when they only use one arm at a time. And it also increases the need for the core to stabilize – a significant benefit, especially with the athletes I'm training.
I normally use this exercise either in the first phase of training, to prepare the body for the heavier barbell overhead work, or during deload weeks.