Records meaning documents, or photographs also? Some compact digital cameras actually have a mode for copying documents. But the overall best choice will be an 8 to 12 megapixel DSLR with a macro lens. That's a real macro lens not a do everying zoom. The macro lens will be able to focus down to 1:1 and is usually the sharpest lens in a manufacturer's line. I use my D200 and 60mm Nikon macro all the time for photographing anying from documents and to paintings. And a flatbed scanner for family photographs.

Absolutely echo Larry's comments. This will give you the best results. It is possible though if a "rough and ready" record of something is needed with minimum fuss to use a bridge type camera to take very close shots, even hand held if you wish. I have used a Fuji S7000 to take pictures as close as 1cm (amazing) and even copy slides and negatives on a lightbox, but obviously the quality suffers. Also, unlike the macro lens with its perfect drawing and flat field, the quick option I have outlined will give masses of distortion (straight lines become curves) and the centre and edge of a document may be difficult to get into focus at the same time (curvature of field). It depends on the quality you need, and in either case you will be able to read the text!

Why not just buy a scanner if you want to digitize family history records. I guess you could use a digi camera as double duty, but it seems that a scanner would be better suited for archiving family records.