Selecting the correct lighting transformer

I'm not sure how to interpret the numbers on my existing transformer in order to select a new transformer.

You just want to match the numbers. It's not that important to understand what they mean. That said, note that the output is 12VAC (AC is "alternating current"). LED drivers are usually DC ("direct current"), but there's this corner of the market that seem to be descendants of the landscape lighting market and low-voltage incandescent lighting market, both of which use AC.

Which means you might have better success finding a replacement transformer by looking for landscape or low-voltage incandescent lighting parts, rather than "LED driver" parts.

So, find a transformer that meets the following:

• 120 volts AC for input (i.e. normal North American household line voltage)
• 12 volts AC for output
• 4 amps (or as the markings show, 4000 mA...same thing) current for output

Be very careful about the output AC vs DC, because this is where you'll likely find the most confusion, since there are those two different styles that otherwise seem very similar.

If you find a transformer rated at higher than 4A, that's fine. You just don't want less. You should try to get 4A or as close to as you can though.

there is a copper wire coming from both the lighting fixture as well as the wiring coming from my ceiling. I understand that this is the ground wire. They were not connected when I pulled the light off. Is this a problem? Must they be connected?

There's a green screw on the ceiling plate, which is intended to be a ground connection. Judging from the bends in the wires in the photos, it looks like the original installer tried to connect the ground wire from the supply cable along with the ground wire from the lamp cover to that single screw. This might even be the way the lamp was intended to be installed, and the wires might even have been connected there at some point, but getting two different wires secured to the same screw like that is difficult.

You would probably have better luck getting a screw nut and a short piece of extra wire (preferably the same size as the supply cable wire, though since they used a smaller wire for the lamp cover anyway, I'd say the size isn't too critical). Use the wire nut to secure all three wires together — the supply ground, the cover plate ground, and the new wire — and then secure the single new wire to the green screw in the ceiling plate. You'll need to straighten out the ground wire from the supply cable.

And yes, it's a good idea to make sure the ground connections are done.

As far as finding specific suppliers, that's technically outside the scope of the Stack Exchange network, including this site. But if you search for "landscape lighting transformer" with your favorite web search engine, you're likely to find a variety of vendors that could be useful. Alternatively, look for parts for low-voltage halogen lighting. These are often used for decorative lighting, cabinets, or track lighting.