The word "Classic" is a derivative of the word "Class" which means a higher standard, something that has standards that others do not have.
Look at the E-Type Jaguar for example it oozes class. Or how about the Porsche 911 with its whale tail.
If the E-Type Jaguar was built to day it would still turn heads.
That's a "Classic Car".
"Vintage" (depending on where you are) usually denotes a car that is at least 30 or 40 years old.
"Antique" is older than that (some jurisdictions have special licences for these types of car).
A "classic" car is generally one whose design, styling or performance was considered exemplary in its day, leading to collectors wating it. Some British classic cars would include the Jaguar XK120 and 150, the early MGs (the MGA for example), the Austin Healey 3000, many models of Bentley, Rolls Royce, Aston Martin, Morgan, the Sunbeam Tiger, etc.