A polite, somewhat formal greeting. Often used when inquiring about someone's well-being or mood. Can be used as a set phrase.
It's been a while. How are you?
How are you? It's nice weather today, isn't it?
お元気ですか is the everyday polite way to ask how someone is, while ご機嫌いかがですか is a more formal, business-or-ceremonial version of the same question.
Polite phrase combining ご (honorific prefix), 機嫌 (mood, health), and いかがですか (how is it?). The expression is a set greeting.