The introduction of high-definition television (HDTV) ushered in an era of much better looking visuals than what we had before. It's a subset of digital TV, and as such, the standards are set by the ATSC.
For comparison, the NTSC standard for analog dictated 525 interlaced lines per screen, although only 480 were actually visible due to a synchronization gap between frames. Standard-definition television (SDTV), even digital ATSC standard, still has 480 interlaced lines. Interlaced means that the odd-numbered lines are drawn from left to right and top to bottom on the screen, and then the even numbered lines are similarly drawn down the screen, until one entire frame has been drawn. There is also enhanced-definition television (EDTV), which displays 480 progressive lines rather than interlaced. Progressive means that each line is drawn sequentially from top to bottom, which makes for a smoother picture, especially during action scenes.
But high-definition television (HDTV) blows these out of the water in terms of picture quality by upping the resolution. Japanese National Broadcasting demonstrated an HDTV system in 1981 that had 1125 lines per screen, more than double that of SDTV. But now, the most common levels of HDTV are 720p, 1080i and 1080p. HDTVs have either 720 or 1080 horizontal lines per screen. The aspect ratio was also changed from the 4:3 of SDTV to the 16:9 of most motion pictures. So 720p has a resolution of 720 by 1280 pixels, and 1080i and p have resolutions of 1080 by 1920 pixels. The "p"' and "i" stand for progressive or interlaced. While progressive is better for motion, interlaced can make static scenes appear to have higher resolution. Things like levels of compression can affect picture quality, but HDTV's images are by and large more crisp, clear and realistic than those from the TVs of yore. Upon their inception, there wasn't much content, but now there are lots of HD broadcasts -- mostly in 720p or 1080i -- as well as Internet content. And Blu-ray (and the now-defunct HD-TV) DVDs are 1080. It's also really hard to find a TV that isn't HDTV in stores.
If that's not enough resolution for you, Ultra-high definition TV (Ultra HD) may be the next big thing. As of early 2013, the two available levels are 3840 by 2160 pixels (sometimes called 4K) and 7680 by 4320 pixels. Both represent a much larger jump in resolution than SDTV to HDTV. As was the case with HDTV, there is little viewing content for them, but that's likely to change over time. They can also be used to view scaled and converted 1080p HDTV content until more is available. But Ultra HD sets are in the $20,000 to $25,000 price range as of spring of 2013, so aren't likely to be widely adopted for a while.