There are several words in English that are used to say that someone is intelligent. They are all synonyms (their denotation or “dictionary definition” is about the same) but have varying and often subtle connotations (shades of meaning or emphasis).
Smart is the common, daily word for someone who learns quickly, has a good memory, and reaches understanding readily.
Intelligent means much the same thing but is more formal and is used more to refer to scholarly (school) learning while smart also applies to non-scholarly smarts. The noun intellect refers to our mind and the noun intellectual refers to a person who has major interest in science and other intellectual—scholarly—knowledge. Scholars are the most common type of intellectual.
Smarts, in particular, street smarts, refers to people who know how to survive and prosper in the real, day-to-day world.
Gifted is a term usually used by teachers to refer to their brightest students.
Bright is another word for smart. It is positive and friendly. (“Oh, what a bright child!”)
Clever has more to do with ingenuity (inventiveness, ability to figure things out or invent new things). Similar words are sharp and quick, which, when applied to a person, mean clever and smart, the latter emphasizing speed of grasping an understanding.
Smart aleck (or, worse, smartass) is derogatory for someone who is “too smart for their britches”). These are derogatory words and somewhat slang-ish.
The expression common sense is used to apply to wisdom we have naturally rather than book learning. “He is very smart, but lacks common sense.”
A geek is slang for a person who is very intelligent but otherwise awkward and clumsy. There is a saying: “Be kind to geeks because someday you will probably work for one.”
Prudent is a synonym for wise with the emphasis on care and caution. The opposite of reckless. Another word for prudent is discrete. This describes someone who knows when to keep a secret and when not to. “Can you keep a secret?” “You may rely on my discretion.”
Finally—and the strongest word available for intelligent—there is wise. Wisdom is considered perhaps the most desirable attribute to have. It goes beyond intelligent, which is the ability to learn and apply easily and rapidly, to a depth of knowledge and understanding. “He is wise beyond his years” is a common expression for a young person who nevertheless seems wise (since wisdom is commonly seen as an attribute that comes with years of experience).
In American folklore, the fox and the crow are seen as smart and clever, and the owl is seen as wise (probably because of its large, fixed eyes—the bird in reality is really quite stupid). If someone is a “wise old owl,” he or she is not just wise but also clever.