Refresh rate is how often a new frame (single image) appears on screen. Because the US electrical grid is 60 Hz (60 cycles per second) most TVs are 60Hz.
All broadcast television signals are 60 Hz, with 1080i having 60 fields (half a frame) per second, and 720p having 60 frames per second. The exception is 1080p/24 from Blu-ray, which is 24 frames per second. In order to display film, which is 24fps, on a 60 Hz display, you need a process called 3:2 pulldown.
LCD TVs exhibit a blurring of fast motion (called "motion blur") at 60 Hz. One of the ways to combat this is by upping the refresh rate. 120Hz TVs, as their name suggests, refresh at twice the speed of regular LCD TVs. There is a slight reduction in motion blur.
240Hz LCD TVs display 240 frames of video per second, and offer a noticeable reduction in motion blur compared to 60 Hz LCDs. 120Hz and 240Hz also have the advantage of being a multiple of 24, so many high-end LCDs with higher refresh numbers also boast a film mode that displays film-based material without the judder-prone 3:2 pulldown.
To create enough frames so that 60Hz video can be displayed on 120 and 240Hz displays, often a process of frame interpolation is used. With video-based sources, this results in a far better image. With film-based sources, the result is often quite undesirable, creating an ultra-smooth effect derisively called the "Soap-opera effect" where all movies have the aesthetic of a soap opera. Most film buffs adamantly dislike this effect. Others don't mind it.
Plasma HDTVs and DLP-based projectors operate on a different principle of creating light, with each frame of video actually comprised of multiple sub-frames. Because of this, they don't need the higher refresh rates of high-end LCD TVs.