Most of the campgrounds at Death Valley are a first-come first-serve basis with the exception of the Furnace Creek Campground, probably the most popular campground choice in the park, which is open all year and reservations are available from Oct to April. Also, be aware that most of the campgrounds are closed in the summer. For more details on the different areas to camp, make sure to check the National Park Service website because they will also let you know if a campground is closed or not. You could also book a campsite (or lodging or RV site) with Panamint Springs which is located near the entrance on the West side of Death Valley.

We often end up planning our trips the week before we go somewhere, so there were no sites available for Furnace Creek when we decided to go. We ended up staying at Sunset Campground (which ended up having plenty of free spots) which is right across from Furnace Creek. It is primarily a big parking lot area for Vans and RV's, but has bathrooms too. We slept in the back of the Jeep. I think I saw a couple of people out there with tents though too.

There are also several great options for people who prefer not to camp, you can read more about them here.