2. Transporter
The transporters were developed for Star Trek out of necessity. Not only did Gene Roddenberry not want to waste time on transporting the Enterprise crew from the ship to a planet each week, but the budget didn't stretch to building a set for a shuttlecraft. In fact, interesting related tidbit The Original Series only got a Shuttlecraft after the producers did a deal with model makers AMT. The company that normally built small plastic model kits would also build a full scale shuttlecraft if they could get the rights to the model kit of it for free. The idea of transportation wasn't new with Star Trek. The first version of The Fly came out in 1958, which featured short distance transportation, and outside of technology the idea of jaunting was introduced in Alfed Bester's 1950s book The Stars My Destination (more recently seen in the 2008 film Jumper). But Star Trek is credited for the transporter, and probably always will be because of the popularity of the show. It has provided the inspiration for scientists attempting to bring the transporter into our real lives. Because we're past the theory stage in some respects. Scientists have used quantum entanglement transportation to fire data some 89 miles between two of the Canary Islands. Don't start getting hopeful, as what this basically did was to send the form of some photons from one island to another, but not the photons themselves. In a world where such information could have been much more easily sent using the internet, there isn't really a future for quantum entanglement on a broad scale on Earth. Not when we can bounce things off satellites in a far easier manner. Probably something that is more easily achievable first is the data storage required to be able to take all the molecular information in the human body and store it digitally. This was actually something that is still a problem in the Star Trek universe as the transporter doesn't do it all at once and in fact processes the information through a buffer at either end. In the Deep Space Nine episode Our Man Bashir, the information related to several crew members fills all the data storage in the station. But considering the speed of evolution of data storage, I think we'll be in a much better position than the Star Trek franchise by that point after all, there is the popular myth that in 1981, Bill Gates said that 640k of memory would be enough for anyone. We've got a lot more storage than that now!