A kill screen is a level in some game that stops the player's progress due to a programming error or design oversight. The game will crash, freeze, or behave so erratically that further play is impossible. The most famous example of this is that from Pac-Man
In many old arcade games, there was no planned "final level" for the game; developers expected people to play the game, die sooner or later, and move on. That is why the games tend to be so hard. However, if someone could (and they have) manage to keep playing for long enough, the game would reach a point where the level counter was so high that it made the level glitch up and become impossible to finish.
Why it happens
While many things could be responsible for the kill screen, usually it is something called an integer overflow. That is when a value in a game (in this case, the level counter) exceeds the available memory. (In an 8-bit game, 255 is the most.) This is what causes the Pac-Man kill screen, because the kill screen level lies on the final integer.
Unfortunately, the era of kill screens effectively came to an end with the introduction of 16 bit games and higher. In these systems, the maximum values are MUCH higher than 255. The maximums follow this equation: 2 to the X power - 1, with X being the number of bits in the name. Using that equation, a 16 bit system can hold 65,535, a 32 bit can hold 4,294,967,295, and a 64 bit can hold a whopping 18,446,744,073,709,551,615. So, it is theoretically possible still, but you now have to get values and integers up into the thousands, millions, or even centillions, eventually. Also, this is only possible in arcade games (Including Duck Hunt, which is an NES game), so linear games or games with final levels have no kill screen level.