While its imperial meaning is historical, 天下 is very much alive in modern Chinese, often used in grand, figurative, or idiomatic ways.
Historical or Grand Ambitions: It's used when discussing history, empires, or large-scale strategies. A founder of a dynasty is said to “conquer the world” (打天下, dǎ tiānxià). A company might talk about its strategy to “win the world” (赢得天下, yíngdé tiānxià), meaning to dominate the global market.
Idiomatic Expressions: It appears in many common idioms. The most famous is probably 天下没有免费的午餐 (tiānxià méiyǒu miǎnfèi de wǔcān), which means “There's no such thing as a free lunch in the world.”
Hyperbolic Compliments: It's often used to mean “in the whole world” to add emphasis. For example, “You are the best mom in the world” (你是天下最好的妈妈).