Digital Television
Digital television is officially the wave of the future: the way all television will be broadcast in the United States as of June 2009. It replaces analog television---a useful and versatile form of broadcasting that served us very well for many decades. Advances in technology have made the old methods obsolete, however, and cleared the way for the new digital format. Understanding how each format works is key to understanding why the change is taking place.
Analog
Earlier analog signals broadcast on a frequency analogous to the information it was sending. The antenna on your TV picked up those frequencies, and you could tune in to a specific frequency by changing the channel on your TV. Anyone who owned a TV before the advent of cable knows that it provided a fairly limited list of options: channels 2 through 13 in VHF and channels 14 through 83 in UHF. That's because analog signals took up a great deal of bandwidth (about 6 MHz per channel), limiting the number of possible stations in any given area. Cable TV provided additional options, but still sent signals in the same analog format.
Digital
Digital TV signals aren't sent in an analogous format. Instead, they break the information down into a series of pulses, each exactly the same frequency and rate as the others. A digital tuner interprets them as a binary code---essentially a series of 1s and 0s, which is the same way your computer stores information. That allows the signal to be much more compact, taking up far less space on the bandwidth and thus being able to carry a great deal more information (as well as freeing up parts of the bandwidth for other uses).
Resolution
Because digital TV signals hold much more data, the onscreen image can be much sharper and crisper. Digital TVs reflect image resolution as the number of horizontal pixels that can fit on the screen and whether the image is interlaced (flickers between one half of the pixels and the other half) or progressive (shows every line of pixels at once). So a digital TV with a 720p resolution has 720 horizontal pixels broadcasting at once, while 1080i has 1080 horizontal pixels flickering back and forth. Older analog TVs displayed their images as about 480i. As of this writing, the best digital TVs are capable of broadcasting in 1080p: exponentially sharper and clearer than the old way.
Digital Converters
All televisions currently sold in the United States are basically digital TVs, though their particulars vary by brand. Older analog TVs needn't be thrown out, however. Digital converter boxes---available at any electronics store for about $50---can interpret the digital signal into an analog format and beam it into your TV. They're very easy to set up and resemble a cable box in many ways. You use them to change the channel instead of your TV and should be able to receive local programming the same way you always did.
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