Well, we are certainly talking about the removable dentures of our nowadays society with its progressive technology, because in the past they were made using very different methods and materials, which greatly influenced the manufacturing and the cost of dentures. Due to lack of technology, most usually the dentures of the past incorporated teeth of passed animals, ivory, bronze or even teeth of deceased people. You can say you are fortunate enough to have born and have lived to these days.
When a patient expresses his want to get a set of dentures (or, what also happens, is when dentists actually tell the patients that the only viable treatment for their problems is getting a set of dentures), a certain interview is done with him to make sure that he understands all the pros and cons of wearing the dentures. You could say that during the interview the dentist also evaluates the health conditions of the patient, but it is very rare for them to find any contraindications against this kind of dental prosthesis.
Of course, the next step is to take an impression of your bite. This is done first outside the dental laboratory, where the mold of your mouth impression gets done. This impression is then given to dental laboratory. Sometimes, only the impression of the lower bite is taken, and if this is the case then the dental laboratory will take some steps to make the impression of the upper bite.
Once both are done, the acrylic material and the wax is applied on the impressions to make the base of the dentures. This pink base is, of course, toothless at first. The lab technician will pick the teeth according to the data that the dentist sends to him (material, color, and etc.). The denture is then polished, cleaned and given to the patient. Of course, the first models are not always 100% precise, so it is very likely for the patients to return in order to readjust the size or the shape of the denture.