Every mathematical object both is and is not a vector, depending on context. To ask whether (3,5,9) is a "vector" is like asking whether Barack Obama is a "member". He is in fact a member of some clubs and not of others. To meaningfully ask "is Barack Obama a member?", you have to have some particular club in mind. To meaningfully ask whether (3,5,9) is a vector, you have to have some particular vector space in mind.
A (real) vector space is a collection of mathematical objects together with some rule for adding two vectors together and some rule for multiplying a scalar (that is, a real number) times a vector. (There are also some axioms these operations must satisfy, which others have listed.) When you have a vector space in mind, the elements of that vector space are called vectors. When you don't have a vector space in mind, those same elements are not called vectors.
Example 1: There is a vector space consisting of all triples of real numbers. You can add two of these, and you can multiply by scalars, in the usual way. When you have that vector space in mind, any triple of real numbers a vector, and nothing else is a vector.
Example 2: There is a vector space consisting of all quadratic polynomials with real coefficients. You can add two of these, and you can multiply by scalars, in the usual way. When you have that vector space in mind, any quadratic polynomial with real coefficients is a vector, and nothing else is a vector.
Example 3 (this is bizarre one, but perfectly legitimate): There is a vector space consisting of all triples of real numbers except for (0,0,0), along with the number 8. You've decided to add these and to multiply by scalars according to the usual rules for triples, together with the rule that v+8=v for any vector v, and r x 8 = 8 for any scalar r. When you have that vector space in mind, (3,7,1), (2,-1,0) and 8 are all vectors, but (0,0,0) is not a vector.
Example 4: Just like example 4, but instead of the number 8, use the unit circle (not the various elements of the unit circle, but the unit circle itself). With this vector space in mind, the unit circle is a vector. With any of Examples 1,2,3 in mind, it is not.
Finally: Take the example from @rob's answer. He is talking about triples for which you have no good definition of addition. (You could try adding two of these triples in the usual way, but you'd get a third triple that isn't one of the allowable triples). If you have this collection of triples in mind, then you do not have any vector space in mind at all. Then (768023, 44126, 295770) is not a vector (and neither is anything else). But if you have example 1 in mind, then the same triple is a vector.