Parachutes are the primary hardware used to recover rockets (with the distinct exception of Falcon 9, and New Shepard). They are lightweight, foldable decelerators usually made of nylon and/or Kevlar, which inflate into a hemispherical/ellipsoidal shaped canopy (or approximate). They usually have several shroud lines (or suspension lines) which distribute the load of the rocket throughout the canopy of the chute.
The canopy is typically made up of gores – which are subsets of fabric that make the chute easier to make. Strengthening elements are often added on gore boundaries.
Parachutes are designed to take a shape with a high Cd (1.5+) to maximize their efficiency. Alternately, parachutes can be designed for high Mach (a ballute), which have been used on Mars deceleration concepts (LDSD). Parafoils are rarely used for rockets, but are another alternate parachute geometry.
See How to: Size Parachutes, How to: Sew a Parachute and the almighty Wikipedia for more.