In order for an argument to succeed in establishing the truth of its conclusion, the conclusion must follow logically from, or be entailed by, the stated premises. That is, if the premises are true, then the conclusion must also be true. An argument with this property is said to be valid. Valid arguments, however, need not necessarily have true premises. Validity only requires that if the premises were true, then the conclusion would follow logically. Below is an example of a valid argument with a false premise:
P1. Either Santa Claus is real or the tooth fairy is real
P2. The tooth fairy is not real
C. Santa Claus is real
This argument is valid because, if both premises were true, the conclusion would follow logically and so would also be true. That is, there is no way for a person to rationally assent to the first two premises, while also denying the conclusion. However, in this instance it is clear that that first premise is false, and therefore the conclusion of the above argument can be rejected even though the argument is valid.
The sort of argument that most people seek to construct, one where the argument is valid and also all the premises are true, is said to be sound. The conclusion of a sound argument is always true, so accepting an argument to be sound is equivalent to assenting to the truth of the conclusion of that argument. Note that a sound argument must also be valid, but a valid argument need not be sound, because one or more of the premises could be false.
What is soundness?: short introduction to the meaning of soundness with simple examples
What is validity?: short introduction to the meaning of validity with simple examples
What is the difference between a sound and valid argument?: some useful answers to this question from mathematics StackExchange
Validity and soundness: an overview of the relationship between these concepts from the Internet Encyclopedia of Philosophy
Truth, validity, and soundness: an introduction with some examples and practise questions