My original tires that came on my lawn tractor were size 15x6.00-6.
I'm not sure what these numbers are (diameter, width, wheel size ?)
Can I use other size tires on my existing wheels?

I'm pretty sure you hit it on the head 15" overall diameter, 6" wide 6"
rim.
If you mow with it, you would want to stay in the ball park to keep the
diameter the same. You could play with width, but it is hard enough to
mount proper sized tires, let alone trying to get a different one to
seat. If it is like most tractors, the wheels become permanently
attached to the axle after a few years and you have to mount the tires
with the wheels still in place.