The bivy, short for bivouac sack, is essentially a tiny, form-fitting tent that looks like a sheltered sleeping bag. Designed with minimal footprint, the bivy folds up small and keeps weight and bulk at a minimum, often weighing under two pounds. Large enough to slip your sleeping bag and yourself into, bivies provide sealed protection from insects and also protect from cold, precipitation and wind. They're designed to be both water/windproof and breathable and are often composed of materials that include Gore Tex and eVent laminates. Bivy Shelters include some form of pole structure to provide extra head room. Bivies can be as expensive if not more than quality backpacking tents and provide much less space and comfort. However, bivies are much lighter and smaller than tents and offer much of the same protection from weather and bugs.
The lightest shelter option available is the tarp. Often made of lightweight sil-nylon, tarps provide a roof over your head and basic protection from precipitation and wind. They are set up using rope and stakes. Tarps do not provide any protection from insects or other creatures as there is no sealing. You can use a ground sheet in conjunction with a tarp to provide insulation and protection from below. Tarps are extremely lightweight, often weighing under a pound, and are also inexpensive in comparison to tents and bivies. They require more strategic set-up than do tents and bivies and are not a good option when bugs are a concern.
Bug shelters employ sealed netting to keep bugs out. They can often be staked with sticks and hiking poles to provide structure. Bug shelters do not protect from rain or wind and are one-dimensional protection. Bug bivies are also available. These shelters are minimalist and lightweight and are a good form of protection for clear weather where bugs are the main issue.
Emergency shelters should never be used as a primary plan for shelter, hence the qualifier "emergency". However, an emergency shelter is an important component of a survival kit and can be used as a last-resort shelter option. Emergency blankets and bivies are common forms of emergency shelters.
Of course, if you want to get all Bear Grylls while backpacking, you build your own shelter. While this isn't an advisable planning strategy, in the event that you become lost in the wilderness, it would pay to have some shelter-building skills. Consider the materials that are available and learn and practice some basic shelter building prior to your camping or backpacking trip. Some natural shelters include the lean-to, teepee and snow cave.