Generally, divergence is a mismatch of the price highs or lows relatively to the highs or lows of an index. It signals the weakness of an already existing trend and a potential reversal.
In an uptrend when price makes to consecutive higher highs but the index makes two consecutive lower highs. This is a signal that the uptrend is weakening. In MS daily chart (below), high 2 is higher than 1, but high 4 in RSI oscillator is lower than 3, so we have a negative divergence.
In a downtrend when price makes to consecutive lower lows but the index makes two consecutive higher lows. This is a signal that the downtrend is weakening. Again in MS daily chart (below), low 2 is lower than 1, but low 4 is higher than 3 in RSI oscillator, so we have a positive divergence.
An alternative of a negative or positive divergence is an oscillator reversal. In the context of an uptrend, we have a negative reversal when the oscillator is reaching a new high but the price fails to do so or making a lower high. This is a sign of the beginning of a new downtrend. A positive reversal is when in a downtrend, the oscillator is reaching a new low but the price fails to do so or making a higher low. This is a sign of a new uptrend.
Oscillators reversals are in a sense the opposite of divergences, although they have the same implications and they are much more rare.