Falling somewhat under the banner of a "Cheese" strategy, the Terran Reaper Rush is aimed at getting out Reapers as fast as possible and rushing your opponent's base to disrupt his economy line by harassing his workers, giving you the economic advantage as you progress into the later stages of the game. In certain cases, the strategy might even possibly win you the game if your opponent is of a significantly lower skill level and did not anticipate a rush due to negligent scouting on his part.
The Terran Reaper is a highly mobile unit and can outrun and outmaneuver many other tier 1 units in Starcraft 2 due to its cliff-jumping abilities and ranged attacks. Despite not being able to leap over direct barriers like enemy force fields or buildings, it nonetheless does not diminish their ability as an excellent scouting and raiding unit. It is unable to hit air units but is highly effective against basic tier 1 units like workers, zealots or zerglings.
This strategy is ideally used against a Protoss enemy as the reaper can outrun a zealot while Zerglings on creep are able to outpace a reaper. Construct both your Barracks and you Refinery at 10 supply and your first supply depot when your supply hits 11. At the same time, upgrade your barracks with a Tech Lab and your Command Centre to an Orbital Command. You can start pumping out Reapers the moment construction of the attached Tech Lab has completed. Speed is of the essence here and depending on the opponent you are up against, you may decide to either send your first reaper to harass once he's out or you can alternatively wait till you have a stable of about 2-3 Reaper before you dispatch them off to your opponent's base.
Once in his base, start focus firing his workers to cripple his economy. In the event he tries to subdue your Reaper by throwing all his workers at you, out micro those by maintaining your distance while making quick stops to return fire. A single reaper can take out 1 worker in just 3 shots which is relatively fast. You would have to retreat once his stalkers are out as they can catch up to a reaper or in the event of a highly successful rush, may effectively win you the game there and then if you additionally took out his offensive force as well as the Pylons powering his gateways, rendering him helpless.
Practice and experience is key here, as you have to decide whether to press on to take the victory or retreat your forces and transit your army into a mid-game build, commencing production of tier 2 or stronger units to add to your overall force.