Given what the character says in the Apology, it seems that Socrates called himself a lover of wisdom (φιλόσοφος) and called his
practice the love of wisdom (φιλοσοφία).
The adjective φιλόσοφος and noun φιλοσοφία, although not unknown, would have been relatively unfamiliar. These words occur rarely in the extant literature prior to Plato's dialogues. The attested occurrences in the 5th century are in Herodotus and Thucydides. In these authors, a lover of wisdom is roughly someone who has an uncommon understanding of certain matters. In Plato, the adjective φιλόσοφος applies much more specifically to someone who is interested in a certain kind of knowledge and who thinks that this knowledge is acquired in a certain way. The wisdom in which he is interested is the knowledge necessary and perhaps also sufficient for living a good life, and he thinks that the method he uses is the one and only way for a human being to acquire this wisdom.
(In the subsequent tradition, there was disagreement about whether the wisdom the lover of wisdom pursues is sufficient for living a good life. In addition, in late antiquity, there was disagreement about whether pursuing wisdom in the way the the lover of wisdom pursues it is the only way for a human being to become wise and hence to live a good life.)
Socrates characterizes the object of his interest as "wisdom and truth and the best state of [the] soul (ψυχή)." This understanding of wisdom as a state of the "soul" (ψυχή) was unusual. Now it is relatively commonplace to understand human beings as psychological beings, beings who act in terms of their "pysche," but this was not always the case. (The word ψυχή transliterates as psyche and is related etymologically to the word 'psychology.') Socrates seems to have understood human beings in this way and to have had a view about how the belief and desire exist in the ψυχή.
The Soul Needs Care
The Clouds of Aristophanes shows that Socrates talked about wisdom as a state of the soul as early as 423 BCE, when Socrates was in his middle forties and Plato was a boy. In the play, Aristophanes uses Strepsiades to make fun of Socrates. He has Strepsiades refer to the denizens of the think-tank that Socrates heads as "ψυχῶν σοφῶν" ("wise souls") who "in speaking of the heavens persuade people that it is an oven, and that it encompasses us, and that we are the embers" and who "teach, if one gives them money, to conquer in speaking, right or wrong" (94-99).
(Aristophanes (446-386 BCE) was an Athenian comic playwright. His Clouds is a brilliant caricature of the historical Socrates and of the new education in 5th century Athens. Anaxagoras introduced the inquiry into nature to Athens. It quickly became an object of suspicion, and Socrates was caught up in the reaction. (Pericles invited Anaxagoras to Athens to become part of his entourage.))
(How the ancient Greeks understood the "soul" (ψυχή) changed significantly over time. In the opening lines of Homer's Iliad, the wrath of Achilles is said to send the souls of heroes to Hades but to leave them, the heroes themselves, as food for birds and dogs on the battlefield. (Similarly, in Homer's Odyssey 11.74, the ψυχή of an unburied hero asks Odysseus to bury "him," where "him" refers to his corpse.) Achilles does not think that the souls he sends to Hades are the people he kills. Nor does he think that the soul is something one could improve or make worse, especially in terms of the "love of wisdom," as Socrates supposed. The soul was thought to be necessary for life. A human being dies when the soul departs from his body, but thinking and feeling in the living is not a matter of this soul. Feeling was thought to somehow depend on the blood around the heart. Homer's "souls" are shades or shadows. They are portrayed as witless and feeble images of the living. This is not to deny that the Greeks in Homer's time thought of human beings and their actions in terms of what we would now call "psychological" states and processes. They did, but they did not think of these states and processes in terms of the "soul" (ψυχή). Just how the word ψυχή came to be used this way is not clear. It seems that over time as it became possible to think of human beings as having a life after death rather than existing as a shade or shadow, it also became natural use ψυχή to refer to this aspect of human beings and to think that this aspect is somehow in the body in the living.)
Socrates seems to have thought that the soul needs care. He thought that human beings are psychological beings and that the "psyche" is in a better or worse state to the extent that it has or lacks wisdom about ethical matters. As psychological beings, human beings do what they do because of the states and processes in their souls. If the soul lacks wisdom and thus is unhealthy, the individual makes poor choices in life. If, on the other hand, the soul has wisdom and thus is healthy, the individual chooses in a way that makes his life as good as possible given the circumstances.
"[H]istorically the decisive step was taken by Socrates in conceiving of human beings as being run by a mind or reason. And the evidence strongly suggests that Socrates did not take a notion of reason which had been there all along and assume, more or less plausibly, that reason as thus conceived, or as somewhat differently conceived, could fulfill the role he envisaged for it, but that he postulated an entity whose precise nature and function was then a matter of considerable philosophical debate.... [W]hat Socrates actually did was take a substantial notion of the soul and then try to understand the soul thus substantially conceived of as a mind or reason. ... By 'a substantial notion of the soul' I [mean]... a notion according to which the soul accounts not only for a human being's being alive, but for its doing whatever it does.... This was not a common conception, it seems, even in Socrates' time, but it was widespread and familiar enough under the influence of nontraditional religious beliefs, reflected, for instance, in Pythagoreanism. And it seems to have been such a substantial notion of the soul which Socrates took and interpreted as consisting in a mind or reason." (Michael Frede, "Introduction" in Rationality in Greek Thought, 19.)
The Question and Answer Method is a way to Care for the Soul
Given the way he seems to try to instill wisdom by forcing his interlocutors to contradict themselves, Socrates seems to commit himself to the view that reason in the human soul consists in more than the ability to make inferences. In addition to this ability, his view seems to be that reason includes the knowledge that constitutes wisdom. Everyone has this knowledge. At the same time, everyone is confused. Their problem is the presence of false belief, never the lack of knowledge, and eliminating inconsistency in belief is the remedy for this confusion. In this way, as Plato portrays him, Socrates seems to think that inconsistency in belief is a sickness in the "soul" (ψυχή) that stands in the way of the good life and "happiness" (εὐδαιμονία).
(The Greek noun εὐδαιμονία is commonly translated as "happiness," but it might also be translated as "well-being." The difficult question is what are the necessary and sufficient conditions for a life to be good for the person living it and thus for the person to be happy. As Plato portrays Socrates, he says that the good life is tied to the condition of the "soul" (ψυχή).)
(The etymological meaning of εὐδαίμων is "blessed with good fortune." The opposite is κακοδαίμων, "ill-fated, ill-starred, miserable.")
This way of understanding Socrates shows why he was doing something important in questioning his fellows in the marketplace and elsewhere. In saying that the soul requires "care" and that this care consists in the love of wisdom, the view seems to be that his "testing" and "examining" helps those with whom he comes in contact to have healthy souls and thus to have wisdom and not be confused in their thinking and behavior. If Socrates is right, then he is a great benefit to the citizens of Athens. He helps them make the right choices and to live good lives.
This way of understanding Socrates also raises questions that assumptions that he seems to make. It is not obvious that everyone has the knowledge they need to live the good life and that their problem is that they are confused because they also have false beliefs. Nor is it obvious that the elimination of inconsistency eliminates these false beliefs.