4K refers to 4K resolution which means that a display device has approximately 4000 pixels of horizontal resolution and 2000 pixels of vertical resolution. It is generally accepted that 4096×2160 represents 4K for Cinema Screens, but other resolutions around these values are commonly called 4K as well.
“Display” is a generic term, and it could be a TV, computer or mobile device, just to cite the most probable ones. Resolution refers to the number of pixels contained within the display matrix within which the pixels are laid out as a 2D grid.
The resolution of 4096×2160 is also referred to as 2160p (2160 “progressive-scan”) and DCI 4K. It is however not along to claim to be “4K”: UHD 4K is also called 2160p but has a resolution of 3840×2160 pixels.
The main difference between the two is the aspect ratio. 4096×2160 is for an aspect ratio of 256:135, while 3840×2160 is for the more common 16:9. In general, it is estimated that UHD 4K (Ultra-High Definition) will be a more common format because most consumer devices such as TVs, laptops and mobiles use 3840×2160.
There are even more “4K” resolutions if you take in to account the cropped formats:
Ultra Wide Television: 5120 × 2160 / 21:9 ratio
DCI 4K (native): 4096 × 2160
DCI 4K Cinemascope (cropped): 4096 × 1716
DCI 4K (cropped): 3996 × 2160
UHD 4K: 3840 × 2160