许愿 can mean a casual birthday wish or a solemn religious vow. In temples, 许愿 implies a promise to return and give thanks (还愿 huányuàn) if the wish comes true.
grammar
Can be split: 许一个愿 (xǔ yī gè yuàn) = 'make a wish.' This verb-object construction allows inserting modifiers between 许 and 愿.
