Can someone tell me what kind of cleat this is? circa 1993. I believe I got it Someone told me that I can just get new shoes (the shoes are completely shot) and reuse the cleat. Is that a good idea or should I get new cleats? I think it's the shoes that are really expensive. I am a very casual rider (go with kids) but I'm now used to the clip in style.
It's a Shimano SPD cleat. There are a few different models, but they are all cross compatible. You can reuse the old ones if you can unscrew it, but new pair should be under 20€, too.
EDIT: Looks like this is an old Shimano cleat, like this one: http://www.fixedgeargallery.com/reviews/dzr/ovis/10.jpg. Modern ones and most copies have a sharp angled "shoulder" on the middle section.
You should accept the answer from OJS but this is too much for a comment.
SPD is the format of the attachment from Shimano. They have a patent on SPD - if it includes the shape of the cleat I don't know. Even if so a patent does not stop copies.
The shoe just has a hole pattern that will accept a number of cleats.
As much as stuff changes in bicycles the SPD has been around for a while.
As beat up as that pedal is I think I would try the existing cleats first. Like I said in a comment try a day of penetrating oil. You could even cut the cleat out of the shoes and soak the whole thing. Not sure if that is worth saving like $13. If you replace the pedals they will come with cleats. Those cleats are not in terrible shape (well compared to the pedal).
And that looks like an original Shimano to me. That looks like the original M737.