Please read the entire essay below for a more coherent picture, but here is a summary I wrote for a friend:
Socrates thought that the unexamined life was not worth living, and that a virtuous life was the best and therefore the happiest life. Being an intellectualist, he was not concerned if other human conditions might affect happiness.
He thought his wisdom depended on the fact that he admitted to being ignorant as to what exactly would lead to such a good life, even though he was considered by Plato and others to have a good soul.
And if he lacked that knowledge – in contrast to the sophists who claimed to possess it and taught it, and whom he convinced through the elenchus (logical refutation) that they too were ignorant – Socrates could not be considered a “teacher” in a strict sense of the word, even if he was perceived like that by people around him.
So what he was primarily involved in was the care of his own soul, and the discussions he held ultimately led the interlocutors – who claimed to be teachers of virtue, etc. – to agree that they too were ignorant about such things.
Nehamas ends by asking, “Or should we return to Socrates’ superficially more modest approach, knowing, however that once we engage in the care of the self we will never know when we can stop, that the limits of the self are also the limits of the world?”
What Did Socrates Teach and to Whom Did He Teach It? – Alexander Nehamas