Ahh, I figured I was explicit, but looking back, it can be a little confusing.
I can give you an exact description though. Think of a steel router bit, with the bottom 1/4" blackened, exactly like a far overdone steak. Just completely black.
Without being able to see the bit, I can only guess from your description that you were using a cheap high speed steel bit (HSS). Oftentimes, unlike carbide bits, there is no cutting part on the bottom of the bit. While it may look like it SHOULD cut, it's not DESIGNED to be a cutting edge. If you notice, on a carbide bit, there will be a chunk of carbide tooled into a cutting edge on the bottom of the bit for this very purpose.
Also, the bottom edge of a HSS bit is a flat piece of metal, even if it will cut, the entire bottom of that bit is running on your material, heating the entire bottom of the bit up. It may work, but it's not the ideal bit for what you're trying to do. As someone stated above, carbide bits are a far better choice.
You've also got to take into account your material you're using. If you were doing solid wood, you probably wouldn't have as big a problem. MDF has binders to keep all the fibers together, heating them up is simply going to add more friction to the bit (which is why they dull HSS bits so much quicker), and plywood, like what you said you were using, has individual sheets glued together. Perhaps you were taking just enough off to hit that glue layer, and were heating that glue up, causing that nice heat buildup, or something similar. Many different things coulda caused your issue.
If you were using HSS bits, which it sounds like you were, then simply throw it away and buy a new one. There's DEFINITELY many reasons they're cheaper than carbide bits, and you happened to find out one of them. It may or may not affect the actual use of your bit, I couldn't tell you without looking at or seeing the bit, but if you had a carbide bit, you could have your bit resharpened and you'd be back in business. With a HSS bit, it's more than likely not meant to be sharpened, and you'll end up spending MORE on them to replace them compared to the life of a carbide bit, but that's an individual choice you'll have to make.