This biomembrane enclosing the vesicle is the same as that of the outer (cellular) membrane. They are a basic tool of the cell to organize its metabolism. Vesicles are used for digestive purposes, as transport vessels, as an enzyme storage, and as chemical reaction chambers. Many vesicles are made in the Golgi apparatus, but also in the endoplasmic reticulum, or are made from parts of the plasma membrane.
Lysosomes (membrane-bound digestive vesicles) can digest macromolecules (break them down to small compounds) that were taken in from the outside of the cell by an endocytic vesicle. This is the basic way for a cell to feed (except for photosynthesis in plants, which don't have lysosomes). The membrane of the lysosome is impermeable for lysozyme, the enzyme that does the actual digestion, to protect the cell interior from being digested by its own enzyme. Lysosomes are made in the Golgi apparatus.
Transport vesicles can move molecules between locations inside the cell, e.g., proteins from the endoplasmic reticulum to the Golgi apparatus, and from there to the outer cell membrane, where they are expelled (exocytosis).
Vesicles can be used as reaction chambers for chemical reactions that could damage the cell if they would occur in the cytosol. For example, peroxisomes are detoxifiers of hydrogen peroxide (H2O2), a toxic byproduct of cell metabolism. Large storage vesicles are known as vacuoles.