'Sleeping under the stars' has a nice ring to it, but it has drawbacks. Unless the weather is perfect and the bugs are non-existent, you'll want the shelter of a tent when you sleep outdoors. There are a lot of tents out there ranging from primitive lean-tos that provide cover from rain, all the way to huge, structured shelters meant to withstand gale-force winds on Everest. Choosing the right tent requires you to consider the type of trips you will be taking (i.e. backpacking vs. car camping), the season and environment in which you'll be camping, and the conditions you'll face.
Bivy sacks and tarps are extremely lightweight, compact shelters preferred by mountaineers, climbers, and minimalist backpackers who need shelter but want to conserve as much space and weight as possible.
For most campers, 3-season tents are the way to go. Intended for relatively mild conditions, these tents have mesh panels to promote airflow and employ rain flies to keep you dry.
Designed primarily for winter, 4-season tents feature heavier fabrics and more poles than 3-season tents. This adds weight but is essential for withstanding fierce winds and heavy snow.