Essentially, an aside is something said by character for the audience's benefit, but it is not meant to be heard by the other characters. Asides are usually a character speaking to themselves, though sometimes characters do actually address the audience in Shakespeare's plays.
An example of an aside is in Act 2, Scene 2 of Shakespeare's Romeo and Juliet. Romeo is outside Juliet's window in the Capulet garden. He responds to something she says, but she cannot hear him yet. He speaks of his love for her and how beautiful she is, which reemphasizes for the audience how much she means to him.