The Classic Car Club of America claims to have invented the term Classic car and thus they believe that the true definition of the term is theirs. According to the CCCA:
This rather exclusive definition of a classic car is by no means universally followed, however, and this is acknowledged by the CCCA: while they still maintain the true definition of 'classic car' is theirs, they generally use terms such as CCCA Classic or Full Classic to avoid confusion.
More common usage, however, fundamentally equates Classic car with the definition of Antique car as used by the Antique Automobile Club of America, who define an Antique car as one over 25 years old. Thus, popular usage is that any car over 25 years old can be called a 'classic car'.
25 years is generally considered a good cut-off age for such terms because it's extremely rare for a vehicle that old to still be owned or used without special consideration for its classic status - by 25 years old, a car will have exceeded its design life by some considerable margin, 10-15 years being the norm barring accidental loss. It will probably need significant maintenance to keep running, and many parts will be hard to obtain through the usual channels. Thus, a non-enthusiast will sensibly conclude that it is not feasible to continue using a car that old for regular driving.
This is not to say that an enthusiast of classic cars might not drive such an old vehicle daily, but that enthusiast will be willing to live with the greater difficulty of so doing.