A function is "into" (i.e., injective or 1-1) if for , and it is onto (i.e., suprajective) if for all there exists .
Read carefully the above definitions, chew them slowly, draw yourself some diagrams and examples to try to fully understands them...and go on with your mathematical life.
Ps. Remember! In mathematics, "to all" means TO ABSOLUTELY ALL, without any exception, and "There exists" means "there exists AT LEAST one (but perhaps some (many) others, too)"...
Nevertheless, Wikipedia and MathWorld, as well as most other places that I can remember call a function with a domain A and a codomain B, "a function from A to B" (notation: f: A -> B). I believe this is the standard terminology. Also, such arbitrary function is said to map elements of A into elements of B, but this is used for elements; I believe it is different from mapping A into B. Wikipedia also has the expression "a function on A into B"; I am not sure about that.
Finally, I don't think I've ever seen "into" mean that the range of the function is a proper subset of the codomain. For example, injection f: A -> B is sometimes called "inclusion", or "embedding of A into B". I am pretty sure that the same terminology would be used if f is surjection as well.
I would also add that inquiring into a definition, especially when it is genuinely unclear, is a legitimate thing. It is quite different from asking for a help with a problem, which may or may not be legitimate.