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The Federation communications earpiece
The cell/mobile phone was not the only aspect of Trek communications technology that made it to reality.
The wireless earpiece, which both communications officer Lt. Uhura and science officer Spock were known to have used on a number of occasions on the show, also became a reality as the Bluetooth Headset, which is also a Motorola innovation.
Motorola Mobility was purchased by Google in August 2011.
Star Trek has had so many computer and information-related technologies showcased in it over the years that it's very hard to tell where the science starts and the fiction ends.
While many of the IT advances in Star Trek haven't made it yet — such as true artificial intelligence, as featured in the Enterprise's shipboard computer system or Lieutenant Commander Data, many aspects of Trek IT have managed to filter their way into our daily lives.
Display technology is probably the biggie.
While you can hardly credit Star Trek for inventing the CRT, you can certainly see how the show has probably inspired generations of engineers to create LCD flat screens and HD widescreen wall-mounted displays, which were simply called "Viewers" or "Viewscreens" on the original show.
It should probably also be mentioned that the telepresence technology using video feeds for ship-to-ship communication first shown in 1966 is now commonplace, using technologies such as Skype, Google Hangout, and Apple FaceTime in addition to corporate video conferencing products such as Microsoft Lync, Citrix GotoMeeting, Cisco WebEx, and Cisco Jabber.
However, it should be noted that social norms have limited their actual adoption, and voice communication, while now more advanced than ever before with the advent of VoIP and digital signal processing, is still the way most people like to do things if they aren't texting or emailing.
My favorite of these display technologies is the PADD, or the Personal Access Display Device. Although the term was coined later in 1987 with the release of the Star Trek: The Next Generation series, it did actually make an appearance a few times in the original show, where it was used by various engineers and administrative staff.
Today, aspects of the PADD can be found in Apple's 9.7-inch and 7-inch iPad, dozens of Android-based tablets, and Windows 8 and Windows RT tablets, such as the Microsoft Surface.