These are all measurements of resolution. A digital television picture is made up of hundreds of lines of visual information, one line stacked on top of another. "Vertical resolution" refers to the total number of visible lines counted from the top of the TV screen to the bottom. Every 720p HDTV has 720 visible lines; every 1080p HDTV has 1,080 visible lines.
The latest advancement, 4K Ultra HD TV, has 2,160 visible lines (double the vertical resolution of 1080p). It also has double the number of lines measured horizontally from edge to edge (3840, vs. 1920 for a 1080p HDTV) — resulting in a picture with four times the total resolution of 1080p. (The "4K" in the name reflects an approximate number of lines.)
When choosing an HDTV, the important thing to remember is the greater the resolution, the greater the picture definition. That's because there are more lines of information, so the level of detail can be that much more precise.
4K Ultra HDTVs represent the highest resolution and, thus, the new standard in picture clarity. 1080p HDTVs are ideal for 1080p content, like Blu-ray movies. For regular TV programming, a 720p HDTV will give you good performance but if you get HD channels you would need a 1080p to take full advantage of that great picture. Many cable providers are also adding 4K channels to their line up for early adopts of 4K.