Something is ancient if it is from the distant past, or is very old. Most things that are described as ancient are from civilizations that have since fallen.
Something is antique if it is old, and people think it is worth collecting. Some antique things aren't worth very much, and people just like to collect them. But other things become more valuable the older they get, and some antique items can cost a lot of money to buy. Antique is often used to describe an old, expensive item, while vintage is often used to describe a less expensive old item, but the two words can be interchanged.
Someone is elderly if they are in their older age. Elderly is a politer way of saying old.
Retro is a style where things that were fashionable many years ago are back in fashion again. A retro style can be achieved by using actual old items, or by making new things that have an old look.