A bigwig is a very important person — in fact, you might call her a VIP. One bigwig at a university is its president, and the main bigwig in state government is typically the governor.
If you're a bigwig in the world of competitive chess, you are a well-known and respected player. A government bigwig is a person with a good deal of power and influence, and a corporate bigwig might own an enormous company. This word dates from the 18th century, when powerful, distinguished men tended to wear large, impressive wigs.