In classical two-valued logic, an argument is said to have validity or to be valid if, and only if, it is the case that, if the premises of the argument are true, then the conclusion must be true. In other words, a valid argument is one where the premises make the conclusion true. There are many other ways to formulate this basic definition: the premises entail the conclusion; it cannot be the case both that the premises are true and the conclusion false; the falsehood of the conclusion entails the falsehood of at least one premise; etc.
A close examination of the definition of 'valid' should make a few things clear about validity. The definition says neither that the premises have to be true nor that the conclusion has to be true. Validity is a conditional notion: what it says is that if the premises happen to be true, then the conclusion has to be true. As far as validity is concerned the premises might be completely and obviously false. Consider an example of a valid argument:
All dogs have eight legs.
The President is a dog.
Therefore, the President has eight legs.
Bear in mind that 'valid' is a technical term in logic: this is a perfectly valid argument. What does that mean, in this example? Something like this: suppose it were true that all dogs had eight legs; and suppose, just suppose, that the President really were a dog; well, in that absurd imaginary world, the President would have to have eight legs. The conclusion has to be true, if the premises are true. So the argument is valid, even though it has false premises, not to mention a false conclusion.
Validity is related but not to be confused with soundness; a sound argument is not only valid, its premises are true as well. All sound arguments are valid but not all valid arguments are sound. Not all valid arguments are valid in the loose and popular sense of this word, meaning 'good': not all valid arguments (valid, as this term is used in logic) are good, or successful, as the above example should show.
Argument form is what makes an argument valid. But a valid argument is one where, if the premises are true, then the conclusion must be true (and here is a way to put it more briefly: the premises make the conclusion necessary). Now put these two propositions together and draw a conclusion:
Form makes an argument valid.
If an argument is valid, then the premises make the conclusion necessary.
Form makes an argument such that the premises make the conclusion necessary.
One can see whether the premises make the conclusion necessary just by looking at the form of the argument. That is why argument form is so important. Look, for example, at the following argument form. In fact, any argument that follows this form is valid. You can see that just by reading it:
All S is P.
a is S.
Therefore, a is P.
Now examine the following argument. It fits that form and is (therefore) valid:
All dogs are canines.
Fido is a dog.
Therefore, Fido is a canine.
In law, validity means the legal existence of a norm. A norm that is authorized by another norm is said to exist in that legal system, i.e., to be valid.