A vegan condom is one that does not contain any animal matter or animal by-products. These contain casein which can make a condom feel smooth but there are other products that can produce the same effect (such as a vegetable binder like thistle extract).
In the historical past condoms were sometimes constructed from woven fabrics (which would be vegan) and lamb intestines (not vegan). Casanova is perhaps the most well known historical figure to use condoms: he had them tailored individually for himself but unfortunately re-used the same one.
While the rubber modern condoms are made from is vegan, casein comes from cow’s milk so hence is not. If you are concerned about whether your condoms are truly vegan check the list of ingredients and see if casein is among them. Some vegan condoms are also produced according to fair trade principles.