Policies regarding the use of cameras as personal copying tools vary widely.
You don’t need a DSLR (although they're great if you have one). A handheld point & shoot works really well, especially when it is 12 megapixels or greater. And your smartphone or ipad might actually be the easiest and most efficient for many researchers.
Common sense / Basic tips:
Make sure you have one or more high-capacity cards for your camera. You need to make sure that you’re getting a high-enough resolution capture for each page so that you can see the details, and that often takes up space. The latest iPhone will give you at most 3264 pixel width, which is generally enough, but does take about 3.5 mb. If you're taking 500 pictures a day at the archives, you would need 8 gb storage space for one week's worth.
To save on card storage space, shoot in grayscale (if color is not important)
Buy an extra battery that can be charging will you're photographing
Disable shutter sound. SLR cameras have physical sound, so you can’t do anything about that, but most camera “click” sounds you can mute and should so that you don’t disrupt the people around you.
Suppress flash and adjust to ISO 1000-1600 for lower lighting conditions
Take a cue from the movie industry: photograph bibliographic or archival information in the first shot, perhaps written on a separate piece of paper, from the title page, or from the archival box front. Another thing some researchers like to do is to write down that information (possibly also including call number and location information), then include that note in each frame.
Turn off auto-rotation. You don't want to have to manually rotate all the pages your camera decides are horizontal when they're really vertical. Trust me, it will make mistakes.
Save a copy of the unedited file, preferably as a tiff, before making any changes.
Advanced features: "Tethered" shooting (sometimes called “remote capture"). Requires special software, often provided with the camera. Sofortbild app for Mac is free.
Policies developed for camera use in special collections facilties generally provide safeguards for issues as:
Let's say you take a picture and it is a 4.5 mb jpg file. If you open it up in Preview (on a Mac), rotate it 90 degrees, then save it without doing anything else, Preview will save it as a 2.3 mb file. That means that even though the pixel dimensions are the same, Preview is recompressing it and throwing out some part of the file. The difference may not be apparent initially, but if you make any other changes to the file, it will get recompressed again and you'll lose even more of the original image.
If you keep a copy of the original file without editing it in any way, you're safeguarding the image you took.