Hegemony (pronounced with a hard 'g') is the power of one group over other groups. Hegemony is mostly used to refer to relationships between different nations (or countries). This might be direct dominance, such as through military might, but may be indirect dominance, such as when a nation can dictate the terms of trade to its advantage.
This word is most often used when talking about international relations. Many recent authors use hegemony to describe America's relationship to the rest of the world.
A hegemonic relationship is usually described as less than an empire, but more than a regional power. This means a hegemon often does not actually take over other nations the way the British Empire did in the past. On the other hand, a global hegemon is more powerful than, say, Iran is in the Middle East.
A hegemon is king or ruler with personal power, not a democratic position.