Every time the space shuttle launches, it ends up in orbit without its solid rocket boosters or external fuel tank. The tank breaks up during reentry and little bits of it land over the Indian Ocean, but the boosters parachute to safety, are picked up by a special ship, and get used over again.
The solid rocket boosters, which together provide over 80% of the liftoff thrust of the space shuttle, expend their 1.1 million pounds of propellant in about two minutes. Since they're just dead weight at that point, the space shuttle drops them, and with the aid of some gigantic parachutes, they land in the Atlantic ocean about 150 miles off the coast of Florida some five minutes later.
Before the boosters get eaten by whales or something, NASA sends out a couple specially-equipped ships to go grab them and bring them back, and NASA has shared the whole procedure on video:
What happens next? Well, if the shuttles weren't all about to be decommissioned and turned into museum pieces, the boosters would be disassembled, re-rounded (the casings squish a little bit when they hit the ocean), packed with solid propellant, put back together, and then reused on another few missions. At this point, it's possible, or even likely, that SRBs (or something very like them) will continue to be used to help launch heavy stuff into space, as is the case with commercial systems like the Ariane 5.
If you want to watch the whole story from start to finish, check out the footage below, which is a complication of video from cameras mounted on an SRB during a shuttle launch.