Most of these tips apply strictly to digital cameras (because that's what I use). If you use a traditional film camera, be aware that some of these apply to you as well, but you might want to ax some of these accessories and instead focus on bringing more film. Even additional types can be handy, like sepia or black and white.
The first consideration is a tripod. Now, I often backpack and walk around when I'm on vacation, and there are options for full-size, or near full-size tripods that are made for that with clever methods of folding and extremely low weights. However, the weight tends to get transferred to the price tag, so it can be cost prohibitive to buy one if you don't use it that often.
That said, if you aren't going far it is a great idea to have a tripod with you. It is much easier to take clean, level shots with a stable base than with just your hands. Even one of the "mini" articulated tripods can be a real lifesaver. These are also great for hikes as they are small and lightweight, and can adjust to fit the terrain.
Another great addition to your bag is a micro fiber cloth. Besides cleaning lenses, you can also wipe off lens caps before putting them back on, remove any buildup from the digital screen, and clean fingerprints off your memory cards should you accidentally touch the copper contacts. A multi-tasking tool that takes up almost no weight or space: a no-brainer.
Speaking of memory cards, bring them. Your camera probably came with a token memory card of 8 or 16 megabytes. Bring it; you never know when you might be able to use just one more picture. The price of memory right now is so cheap, if you were going to by an 8 gigabyte card, why not add on another 2 gigabyte, or even purchase two 8 gig. cards? The greatest advantage of a digital camera is "unlimited photos" so take three of everything and decided later which ones to keep.
Another benefit of having multiple memory cards is that you can split them up. In the trip I mentioned earlier I also visited London and Berlin. Having multiple cards can allow you to take pictures of one place on one card and then put it in your luggage for later. This also helps you to avoid a total loss if something terrible happens and your camera is lost or stolen.
Since we've covered storage, let's talk about the other major requirement of digital cameras: power. It is a great idea to at least carry spare battery pack with you (or change of batteries, if your camera doesn't have its own pack). On extended outings, it may also be worth it to carry a charger for the battery pack. This way you can guarantee you'll never miss that one shot that could lead to an awesome photo.
If you keep a few of these accessories in your camera bag, you're almost guaranteed to get all the best photos you possibly can.