It all depends on how you define important, of course. But to my mind the most important invention is telecommunications technology: the telegraph, the telephone, and now things like the Internet. Until about 150 years ago, it was impossible to communicate with someone in real time unless they were in the same room. The only options were to send a message (or go in person) by horse or ship.
The early optical telegraphs of the 1790s made long-distance communication possible at hitherto impossible speeds, at least for the governments that built them, but they were not available for general use. Then in the 1840s, the electric telegraph enabled people to send messages over great distances very quickly. This was a step change, though its social consequences took a while to percolate. At first, telegraph operators became the pioneers of a new frontier: they could gather in what we would today call chat rooms, play games over the wires, and so on. (There were several telegraphic romances and weddings.) The general public, of course, was still excluded, and had no direct access to the real-time nature of the technology. But the invention of the telephone in the 1870s made real-time telecommunications far more widely available.
Today, in the developed world at least, we think nothing of talking with people on the other side of the world. During the course of a normal working day, many people spend more time dealing with people remotely than they do face-to-face. The ubiquity of telecommunications technology has become deeply embedded in our culture. Of course, life has sped up as a result. But we watch TV and use telephones, fax machines and, increasingly, the Internet, almost unthinkingly. If the mark of an advanced technology is that it is indistinguishable from magic, then the mark of an important one is that it becomes invisible — that we fail to notice when we are using it. That makes the significance of telecommunications technology very easy to overlook, and underestimate