Hiking in the desert isn't really all the different from hiking anywhere else, but you need to bring a different set of equipment.
  • Water. I can't emphasize that enough- bring lots and lots of water. You're going to need a gallon every day. This limits how long you can stay out without resupply, unfortunately, as water weighs about 8 pounds per gallon. You can get by on less in the short-term, but I don't recommend it.
  • Food. You won't need as much of it, but you don't want to bring anything salty or exceedingly dry.
  • Clothing. You want light clothing and a hat. You either need to wear long-sleeved shirts and pants or sunblock, because sunburn is BAD, mmmkay? The hat needs to provide you with a reasonable amount of shade- no fez! You want clothing that has a loose weave, and light-colored clothing is better than dark. White is best, but a lot of people don't like wearing white shorts. Bring a jacket too- the desert cools down quickly at night.
  • Map and compass. This is much more important in the desert than anywhere else. Most places in the United States, you can't walk twenty feet without tripping over an interstate, but in the desert it's much easier to get lost and much harder to find your way back to civilization.
  • Boots and sandals. These are the ideal footwear- you want to wear boots while hiking, because you need support for your ankles, but you don't want to have to wear confining boots around camp.
  • Flashlight. Not as important as elsewhere, but still a crucial item. On the full moon, it's not much of an issue, but on a moonless night, you'll thank me.
There's not much else you really need to know. Despite people's worries about rattlesnakes, they'll ignore you if you ignore them, and they're pretty rare to begin with. Ditto scorpions. In fact, in terms of dangerous animals, you're safer in the desert than anyplace else.

Really, the desert ain't bad as long as you don't get lost and carry enough water. If you do get lost, find shelter and do your best to let people know where you are. YOU ARE LEAST LIKELY TO DIE BY JUST SITTING STILL AND WAITING. Moving around is a bad idea if you're lost.

The most likely health problem you'll have while hiking in the desert is heat exhaustion. To combat this, drink plenty of water (you're drinking enough if your urine is still clear. Yellow is bad.), stay in the shade, and do most of the common sense stuff.

The most important thing is to make sure someone who cares about you knows where you are.

Really, most of this stuff is common sense.