What is meant by center of someone's universe?
I read this sentence:
Her young son is the center of her universe.
But I couldn't figure out the meaning.
By convention, a "metaphoric" usage such as his universe refers to the universe as perceived by him (that's to say, everything he's aware of and/or cares about).
By convention and implication (both metaphorically and according to Einstein's Theory of Relativity), every individual is "at the centre" of their own universe. So it's just piling one more metaphoric allusion on top to say that something other than one's own self is at the centre of one's universe.
In short, He is the centre of her universe means she thinks he is the most important thing in existence. A more "folksy / biological" (as opposed to "material / scientific") expression would be He is the apple of her eye.
This usually means a person or a thing that they care about a lot, even more than anything else. If someone is the center of my universe they mean a lot to me. I think about them all the time, want to spend time with them, and focus a lot of my time and efforts on them.
This is a metaphor! That means you're comparing two things without using the words "like" or "as". In this case, you compared a person to the universe. You can also compare other things, like a really tall thing to a tree. "The chair was like a tree, towering above the table."