Your age is the number of years you've been alive. If your cat is six, you can say that he's "six years of age."
You can also talk about a stage of life or an era in history as an age: "School age kids need to get outside and play every day." Often when someone mentions age as a concept or idea, they mean old age, or being elderly. And the verb age means "grow older." The word comes from the Old French aage, "age or lifetime," from the Latin root aevum, "lifetime, eternity, or age."