Our civilization is heavily dependent on communication. Let's take you reading this blog. I had to write the content. It goes to Google's servers, which stores it. You request it. Google's server has to send it to your computer, which displays it in a form that you can read. Chances are, the connections were not direct, since Google and I are a very significant distance apart, about 2,000 miles by my estimation. You might be closer to them, or farther.
But even if you're utterly antipodal to Google, they can send you the information fairly quickly. Extensive fiber-optic lines have been installed between every continent on Earth. Fiber-optic cables send data by light beam, which of course travels at light speed. Data arrives anywhere on earth in less than a second. Data would reach the moon in 1 second. If you have older, copper-based lines, they're slightly slower, merely 80% of the speed of light, but still fast enough to reach anywhere on earth in one second.
But I bring up the moon for a good reason: Several nations now are considering building a colony on the moon. Doing so would bring them massive technological prestige, the "Hey guys, check out what I can do, huh?" factor. Communication will of course be essential to the success of the colony, since the sponsoring government will want pictures and video to wave in the faces of their rivals, and the colonists will probably want news, email, and to call their earthbound loved ones. A radio-link could be established without too much trouble, and a 1 second delay isn't too agonizing, although it will mean lots of awkward silence in phone calls due to the other person not hearing you for 2 seconds at a time.
But what if we then go on to colonize Mars? A Martian colony would be an even-greater version of a lunar colony, but would have even greater difficulties. Wikipedia's army of science nerds has already determined that the delay on Mars would be at least 3 minutes (when Earth and Mars are closest), up to 22 minutes when they are at their most distant. Double this time for anything in which a reply is expected. The delay is so great that a telephone call would be for all intents and purposes impossible. Even radio-backed Internet solution would only be possible when the two bodies are nearby, as almost all earthly computers cannot accept a 44 minute ping time. (12 minutes is the longest trip I found documentation for.)
This may be simply the hard laws of physics working against us. Any distant outpost of humankind may simply be out of touch with Earth forever for all practical purposes. I'd like to think that some faster communication means may be established, such as quantum-entanglement, but there's no room for wishful thinking in engineering.