Rockets will typically have multiple "stages." The first stage is a big rocket engine and burns long enough to escape the atmosphere. Normally, this first stage engine is then detached from the top part of the rocket carrying the payload, burning up in the atmosphere or crashing into the ocean. The next stage(s) continues to burn to make the rocket go faster until it gets into orbit. Once it is in orbit, the payload can be put into orbit. This can be delivering astronauts or supplies to the International Space Station, placing satellites into orbit, or performing experiments.
As mentioned, the first stage engines burn up in the atmosphere or crash into the oceans. SpaceX came up with a genius plan to save just enough fuel in the first stage to land it back on Earth. This way, it can be reused for a following mission, reducing the cost of the launch by roughly $50 million by some estimates. SpaceX claims that the first stage rocket accounts for 75% of the cost of each launch.
This has been very difficult to do because the design is not meant to land, it is meant to launch. It is very tall and top heavy. It is like dropping a pencil from the top of Empire State building and hoping it lands upright. Previously, SpaceX had been attempting this at sea using a floating landing pad because they were unsure of how accurately and safely they could return the rocket. Although they were unsuccessful in landing, they were able to demonstrate that they had some degree of accuracy. This time they used a landing pad on land, making it easier to recover the first stage.
This may be a catalyst for space travel, allowing relatively cheap launches into space. This allows companies to launch more satellites, scientists to run more experiments, and to put more people in space.