Medium-Firm (aka Firm) Tofu has been aged and pressed a bit, and consequently will be more solid (as the name implies) than some other types of tofu which aren’t.
To make it, soy milk with a coagulant is poured into a vessel or mold. A cotton cloth is placed over top. When the bean cured has solidified, weight is pressed on top to force some water out.
Owing to its lower water content, firm tofu is more solid than silken tofu, has a coarser texture, and retains its shape better when cut. When sliced, it almost appears to have a “curd-like” aspect to its texture inside.
One of its names, “cotton tofu”, comes from its surface texture, which shows the weave of the cotton that the tofu is wrapped in to press the water out of it.
There are softer tofus (soft and silken); there is also a firmer one, called “Extra-firm tofu.” Some makers may label their medium-firm tofu as “medium-firm tofu”, to show its place in between the soft and extra-firm types in their product line-up; others may just call it “firm.” Some may even make a medium-firm, firm, and extra-firm, but the medium-firm and firm will be much the same, differing only in one being pressed just a bit more than the other.
If you want to cube tofu for frying, this firm / medium-firm tofu — or extra-firm — is the kind to use.
Unless you have a recipe to follow that directs otherwise, marinate the cubes for an hour or so before cooking in a blend of soy sauce, flavored vinegar, oyster sauce, etc. Then bake or fry sprayed with cooking spray, or in oil.
Compared to softer tofus, medium-firm tofu will be slightly higher in calories because per gram, you’re actually getting more tofu and less water.
Compared to extra-firm tofus, it will be lower in calories and depending on the brand, may even have a fat content that is half that of the extra-firm.