Power spikes are moments in the design of the game where one player/character is really powerful relative to another player / character. Sometimes in League of Legends heroes are designed to be kind of weak at the start or strong at the end. Sometimes a character is just really strong relative to others for a period of time because you unlock a power before the counters are unlocked, or you purchase a powerful item.

Power spikes can be fun because it can be fun to have a feeling of power, or see the balance change in your favor in a game. They can be frustrating because you can be faced with a power spiked player and feel helpless. It is a design choice with pros and cons.