Someone who's elderly is very old. Your elderly neighbor might need help shoveling snow from her sidewalk and going grocery shopping each week.
When you talk about an elderly person, you might call him a "senior citizen" or an "older gentleman." We tend to think of people as elderly when they're quite aged and have started to become physically frail. You can also use the adjective for animals or even inanimate objects that are getting old: "That elderly oak tree is going to have to come down." The Old English root ealdorlic means "princely" or "excellent."