Consider a game where everyone in a classroom picks a number between 0 and 100. The person closest to half the average wins.
Obviously, picking a number over 50 would be silly. Based on this, picking a number over 25 would be silly. Similarly, picking a number over 12.5 would be silly. Continuing this line of thought, picking any number other than 0 would be silly. These thoughts show the level of thinking of those involved, from level-1 to level-.
Level-k thinking analyzes such games. In actual experiments, players never pick 0, which would be suggested by a Nash equilibrium. Instead, they utilize many different levels of thinking.