Overdentures are usually made when there are few teeth left, or the teeth remaining are in poor position to retain a partial denture, or when all teeth are gone from either the upper or lower jaw.
From a practical standpoint, most dentures are full dentures that fit OVER either the retained roots of a few remaining teeth or (these days, more frequently) implants. In the case where a few teeth are remaining, the teeth must have root canal treatment performed. Then an part of something called an "attachment" is placed in the root. The other side goes on the corresponding undersurface of a denture. It is useful to think of a snap on a garment as an analogy. The overdenture snaps into the attachment in the root (or implant), holding the overdenture in place.
The vast majority of overdentures are made as full lower dentures. There are a few lucky patients who function reasonably well with conventional full lower dentures, but most people tolerate them at best. There is almost never any "suction" and the movement of the tongue, lips and cheeks often tend to dislodge the denture. Adding a couple of overdenture attachments can vastly improve the retention and function of a full lower denture.
When an upper overdenture is made, the most common reason is for the patient who cannot wear a full upper denture without gagging. Before implants were widely used, dentists would struggle with gagging patients. We would sometimes shorten the rear edge of the denture. This would greatly compromise the suction and retention of these dentures, making it more similar to the situation with most lower dentures. Overdenture attachments can make it possible to remove some of the palate of the denture and retain good functional retention.
Implants and overdenture attachments are not cheap; but they are certainly far less costly and involved than trying to restore a full arch of teeth with individual implant-retained crowns. In many cases, especially for patients who have lost their teeth due to periodontal disease, fixed implant restorations are not an option, due to inadequate amounts of bone in the required areas. But overdentures are a nice compromise that can turn the struggling denture wearer into a satisfied patient.