What have I got here? Jens Risom Design desk with return  

Those drawer pulls will be a dead giveaway to someone more knowledgeable than me. Just a heads up, be careful asking about value. It's against forum rules. Just FYI
Josh

Definitely Risom. Google "Jens Risom partners desk", you'll see your exact desk. It's nice, I hope you keep it!

I do plan on keeping it for my kids, I grew up sitting at it playing with a typewriter & later a computer.
Sorry about asking value, just wanted to know if it was something unusual or what & whether I should refinish it.
It's in pretty good shape, but could use a cleaning up for sure. How do I refinish or touch it up without wrecking it? It looks oiled. I know all my dad ever did to it was put lemon oil on every once in awhile.

Also, how can I figure out what kind of wood it's made of?

It's definitely a walnut veneer if not solid walnut. Wood grains can be pretty distinct if you keep an eye out for the differing elements. A lot of Danish furniture is Walnut or Teak - Maple was also a really popular look in the Mid 20th century.

It's solid walnut, no question.

"Solid" in this context actually tends to mean solid wood, as opposed to veneer over substrate. And the desk is generally what you would see referred to as a "veneer desk" because the top surface is walnut veneer. the top has solid walnut edge banding. The legs are solid walnut. The sides of the drawer cases are veneer.

I'm pretty sure the top is solid, if you look underneath, the unfinished bottom has the same grain as the top, and it's at least an inch thick.
So just oil it? Should I Google walnut care?

The top is veneered as are all the panels and drawer fronts.
Rub the whole thing down with mineral spirits or naphtha, clean rags, elbow grease, and proper safety precautions. Re-oil with whichever name-brand "Danish" furniture oil finish that strikes your fancy following instructions printed on the container. After your final application of oil finish has fully cured (a week), a paste wax can be applied to even out sheen and add some protection.
The lighter-colored area on the top can't easily be made to match. I'd leave it as honest evidence of the history of the desk.
Nice family piece.

Once you've looked at enough veneer you can spot it just based on the appearance of the grain. When in doubt though, it is good to look underneath and see if it looks the same. This is not fool proof though, as all substrates have to be veneered on both sides with similar or the same species veneer to keep the substrate from warping. So when you see walnut on the top, and walnut underneath, you have to make sure you are not looking at veneer in both places.
In this case one great indication that you are seeing veneer is the edge banding (side grain) where there should be end grain were it a solid board. You can't put edge banding on the end grain side of a solid wood table top, not if you expect it to remain glued for very long. The solid wood top will expand or contract with changing humidity, and break the glue bond.
Nothing wrong with a veneer piece. Veneer has benefits that solid wood does not. And vice versa. Just don't ever go at it with a sander thinking it is solid. You will be bitterly disappointed.