Formal & Political Context: This is the most common use of 伟大. It appears constantly in government speeches, news reports, history books, and national slogans to describe leaders, the Party, the army, or major national projects. It carries a tone of utmost respect and solemnity.
Describing People: When used for a person, it implies they are a historical figure, a national hero, or someone who has made a profound and selfless contribution. A doctor who sacrifices their life to save others could be called 伟大. You would not use it for a successful but ordinary CEO.
Describing Concepts: It is often used to describe profound, abstract concepts like love, sacrifice, or dreams. The most famous example is 伟大的母爱 (wěidà de mǔ'ài), “the great love of a mother,” which is seen as selfless and powerful.
Sarcastic or Casual Overstatement: Among younger people on social media, 伟大 can be used sarcastically to describe a trivial accomplishment, precisely because the word is so formal. For example, “He finally washed the dishes after two weeks. A truly 伟大 (wěidà) achievement!” This usage is informal and relies on the listener understanding the humorous exaggeration.