Short answers, pulled from the story.
Trans-lunar injection is a large propulsive maneuver that uses a chemical rocket engine to increase velocity significantly. This burn changes the spacecraft path from a circular parking orbit into a highly eccentric ellipse with an apogee near lunar distance.
The Soviet Union launched Luna 1 on the 2nd of January 1959 as the first probe attempting trans-lunar injection. That mission missed the target by more than three times the Moon's radius and sent the craft into heliocentric orbit instead.
Japan launched Hiten in 1990 to explore novel low delta-v transfer methods taking six months instead of three days. SMART-1 used solar-powered ion engines for propulsion starting in 2003 and took over thirteen months to reach lunar orbit.
Engineers designed specific TLI burns to target free return trajectories during early human spaceflight programs such paths allow a spacecraft to loop behind the Moon and return to Earth without further propulsion. A failure in the main engine would still guarantee a return home under gravity alone.
Each burn lasted approximately 350 seconds while providing velocity changes between 3.05 and 3.25 kilometers per second. The spacecraft reached speeds near 10.4 kilometers per second relative to Earth after the engine cutoff.