Divergence is when the price of a security and an indicator or oscillator, move in opposite directions of each other. One gains while the other loses. So simply put you have two lines that are basically moving drastically apart from each other, instead of moving together in similar patterns. This often signals a potential pending change, stall or shift in price movement. When doing your chart technical analysis, divergence is either negative or positive depending on the separation or move.
Negative divergence happens when the price of the security makes a new high, but the indicator fails to do the same and instead closes lower than the previous high. Positive divergence occurs when the price of a security makes a new low while the indicator starts to climb upward.
Most often traders will note key divergences occur between candles and a MACD Histogram, an RSI (Relative Strength Index), a MFI (Money Flow Index), a Stochastic indicator, an Elliot Wave index or other similar indicator that notates the volume or activity of trading & interests relative to price.
Chart Example from stockcharts.com