The plot of A New Hope revolves around the Death Star technical plans, which Princess Leia gives to R2-D2.
How did she and the rebels get hold of the plans?
The short answer is that we don't know (yet). The mission to steal the plans will apparently be a major part of the new Star Wars spin-off film; Rogue One.
"Edwards revealed that the film will revolve around a rogue band of resistance fighters who unite for a daring mission to steal the Death Star plans and bring new hope to the galaxy. “I couldn’t sit in a cinema and know that someone else made this film,” Edwards said in the panel. Fans were treated to concept art and a brief video that caused a packed house to erupt."
As for the EU version(s) of events, this has now been completely disregarded. Pablo Hidalgo famously quipped;
"...if you had to throw a dinner party and invite everyone who had ever stolen the Death Star plans, you'd be surprised at how many place settings you'd have to worry about."
In the X-Wing video game series, you get to participate in several parts of Operation Skyhook, participating in several missions to gather data, cover the theft of the plans and help get the plans to Princess Leia aboard the Tantive IV.
In the spoof comic Tag and Bink Mon Mothma's lover is a man (if I recall correctly) named Manny Both Hands. He was a spy who stole the plans in a single handed act of espionage. However Tag and Bink accidentally cause his untimely demise.