Recently, a brief discussion of tikkun olam occurred on an online Gnostic group that I frequently read but rarely reply to. As I was not part of the original conversation I refrained from discussing specific questions that were addressed to someone else. That's only polite, right? I do have some thoughts on the subject however, and as I haven't blathered on about kabbalah for a while it might be time for a change of topic.
Tikkun olam is usually translated as the World of Rectification. Olam is an interested word. It certainly can mean world, but can also mean eternity and may be definitionally, though not etymologically, related to the Greek word "kosmos" in its Neoplatonic sense. Tikkun means rectification, which is to say to restore something to its proper condition. To fix it so it was the way it used to be.
The idea pre-dates kabbalah and is directly related to Jewish messianic thought. While pre-kabbalah in nature, it is still a part of kabbalistic tradition, found in the Zohar and heavily developed by both R. Moshe Cordovero (the Ramak) and R. Titzchak Luria (the Ari) and has so become an exceedingly important element of kabbalistic thought and practice.
Generally speaking, we may talk about tikkun olam as a goal and tikkun a the process for achieving that goal. But what is tikkun olam, the rectified world? That is a difficult question to answer, and Jewish thought varies, typically along two lines, either physical or spiritual, or, of course, both. In either case, tikkun olam might be thought of as an equivalent to the Christian idea(s) of the Kingdom of Heaven/God as well as the kabbalistic idea of the World to Come.
So, some kabbalistic background: The Zohar introduces an idea, a myth, about the creation of the world (olam) through a process involving too much Gevurah. The problem isn't that creation occurred through Gevurah, almost all creation in the Zohar comes through the Gevuric process. The problem was the amount of Gevurah involved. The great and terrible force of Pachad shattered the vessels made to contain the divine light and everything below a certain level fell, resulting in the development of the physical realm, which wasn't necessarily supposed to happen. (This is a difficult issue in and of itself. God, being perfect and all knowing, would have known this would be the result of the action, so presumably the physical realm was intended, but it could not, should not, have been part of the process of emanation, it had to be created instead).
Tikkun is the process of picking up the shattered pieces, the kellipot or shells, and putting their contents, bits of divine light, back where they belong. In the Zohar the kellipot are demonic beings, or even beings at all, but appear as coverings for the bits of divine light in the physical and sub-physical realms, distorting the perfect light in a way not dissimilar to the later Neoplatonic idea of hyle and its affect on the vehicle of the soul.
The personified evil forces in Zoharic thought are not described or called kellipot but are instead the forces of the "Left-Hand Side," Samael and the Evil Inclination and their ilk. Even they are not evil in the classical, opposite of good, way. When properly directed the Evil Inclination is the source of artistic creativity and the drive towards procreation and Samael punctures the over-inflated egos of the faux-righteous. In kabbalistic thought evil is as complex and necessary as good. It is only when evil, or good for that matter, is improperly directed that problems occur. (Compare this with Pseudo-Dionysius' writings on the subject of Evil, which are derived from Proclus and Plotinus).
So, how do we go about practicing tikkun, the ultimate form of kabbalistic theurgy? That is, of course, complicated. One method is largely mundane: be a good person and do good things. An orthodox Jewish conception of this is to follow the commandants of Torah. Kabbalistically, the answer is essentially the same, except the mitzvot, commandants, are understood to have not only earthly ramifications, but entirely metaphysical ones as well, ones that affect not only soul, something found in less mystical interpretations of Rabbinic law, but also the entire spiritual structure of creation.
Beyond this, Jewish mysticism has a long-standing form of meditation known as hitbodedut (see, for example, http://jwmt.org/v2n20/hitbodedut.html, see especially the section on hitbodedut and tikkun!), which appears to have its roots in merkavah mysticism, but was also influenced by Sufism. Hitbodedut involves the mental, and presumably spiritual, manipulation of the letters of the Hebrew alefbet. The alefbet is considered to not simply represent, but to spiritually BE the divine forces God used to create. By manipulating the letters, reality can also be altered. Hitbodedut is a theurgic process, perhaps similar to the Neoplatonist Iamblichus' idea of hieratic prayer, which involves the use of "nonsense" sounds that have meaning not to us but to the gods.
I'm not really going anywhere with this, mostly thinking aloud. That being said, the ideological connections between Neoplatonic theurgy and kabbalistic tikkun should be clear. Both are an engagement with the spiritual realm for the purpose of moving the physical realm towards its divine purpose. In doing this not only is the physical realm rectified to the divine plan (or whatever), but so is the individual soul, which is, after all, a microcosm of the macrocosm.
As I am involved in the development of an ecumenical, Neoplatonic gathering (Ekkelsia) that, amongst other things, will focus on a kind of liturgical theurgy, there might be more to say on the subject. But that will have to wait.