There are lots of great backpacking areas in Nevada. A lot of them depend on the season and water sources. In the summer we all head to the mountains and the other seasons we venture into the desert. Since I live in Southern Nevada I can give you a few good ideas on backpacking around Las Vegas.
In the warmer months MT. Charleston or the Spring Mountain NRA is the place to go. There are lots of great backpacking trails around the MT. Charleston areas. The MT. Charleston peak trail is about 18 miles RT and goes up to the 11,919 foot peak. The Bonanza Trail is a good weekend trip but requires a car shuttle.
Red Rock Canyon is a great day hiking area and it does get a lot of day traffic since its really close to Las Vegas. You can backpack at Red Rock but you have to camp at 5,500 feet or higher which puts on up high or on the backside of Red Rock.
The Lake Mead NRA has a ton of great areas to explore as Nick has stated. You might have to carry all of your water depending on your location you select. There are some established trails but mostly you pick an area and go for it. Bowl of Fire, Pinto Valley and some trails leading down to the Colorado River are really nice trips.
Valley of Fire State Park is a great place to visit and has some shorter day hikes. Backpacking is not really allowed because of the lack of longer trails and no water sources.
The Muddy Mountains Wilderness area near Lake Mead and Valley of Fire is one of the best desert backpacking areas in Southern Nevada. Lots of beautiful scenery, solitude, and some amazing prehistoric rock art if you can find it.
The Gold Butte area is pretty amazing. Most of the trips there you would have to look at a topo and just go. Don't forget its pretty remote and carry lots of water.
I have limited backpacking trips throughout the rest of the state. The Ruby Crest Trail near Elko is probably the best longer trip in the state. The trail is about 43 miles long and almost every night you can camp at a beautiful alpine lake. Part of the Tahoe Crest Trails runs along the Nevada portion of Lake Tahoe.
I hope you are able to have some amazing trips. If you have any questions PM me.