The party was the perfect opportunity for everyone to let their hair down and dance like nobody was watching.

let 是动词，意思是"允许、让"。 their 是代词，表示"他们的"。 hair 是名词，意思是"头发"。 down 是副词，表示"向下"。 let one's hair down 是一个习语，字面意思是"把头发放下来"，引申为"放松、不拘束"的意思。这个表达来源于过去女性在正式场合要把头发盘起来，只有在放松时才会把头发放下来。