A mathematician would tell you that a vector space is a commutative group (whose elements are called vectors) combined with a field (whose elements are called scalars) where a few natural axioms apply. You don't need to know the exact definition but you have to keep in mind that it is a very broad definition that works in many different cases.
You can add and subtract vectors and you can multiply them with scalars. If you do both at the same time it's called a linear combination. If b1, ..., bn are n vectors and x1, ..., xn. Then the vector a = x1b1 + ... + xnbn is called a linear combination of them .
You can apply a function to vector and make a different vector out of it. An important case is that of a linear function (or linear transformation). f is a linear transformation if f(xa + yb) = xf(a) + yf(b). In finite spaces linear transformations can be represented by matrices.
For example symmetry operations are linear transformations . The most important linear transformation for a chemist is the Hamilton operator . Pretty much every operator a theoretical chemist uses is linear. When dealing with vector spaces of functions  people like to say linear operator instead of linear transformations but it is the same. For example eigenvalue theory is alike. Then it does not surprise us that the Schrödinger equation can be reduced to a matrix eigenvalue problem according to the Ritz method or Hückel method.
 By "clean" I mean avoiding calculus. Calculus may be useful but it seems like a witchcraft to me.
 Vectors are bold, scalars are italic. You could write vectors with little arrows but it gives the wrong impression.
 A special kind called unitarian but you need a scalar product to say that and that's not today's topic.
 Another special kind called hermitian, you need a scalar product again.
 The word function is very ambiguous since everything we are dealing here is function. Even a tuple (which is normally called vector by physicists or chemists) is a function. So we should specify that we are talking about functions whose domain are the real numbers.