Haraë is a central concept and ritual in Shinto. It is normally translated as “purification”, and this is not a bad translation; haraë is how one gets rid of kegarë, or impurity. Haraë is very closely linked to misogi, which is also a way to get rid of kegarë. Indeed, in contemporary Shinto it is not clear that they are really different, and it is not uncommon to see references to “misogiharaë” or “haraëmisogi”. The main difference is that misogi involves water, and haraë does not. I have written a whole essay on this topic for my Patreon, which is available as a back issue if you are interested in more details.
When one enters a jinja, one is supposed to rinse one’s hands and mouth at the water basin near the entrance. This is an abbreviated form of misogi. On the other hand, if you participate in a matsuri, the priest will wave an ōnusa over you at the beginning, after reciting a haraë norito. An ōnusa is a stick with many strips of paper, linen, or thin rice-straw rope attached to it. This is a simple form of haraë.
In haraë, it is common for the kegarë to be symbolically transferred to another object, which is then disposed of. For example, it is common to transfer kegarë to a small piece of paper, cut into a roughly human shape to make a doll, by rubbing it on your body and then blowing on it. The dolls are then either burned or thrown into a river to carry the kegarë away. At some jinja, people unwind a small piece of rope as part of the haraë, and then the remains are burned or washed away. The idea here is that, as the rope is unwound, so too are the binding effects of the kegarë on the person. The ōnusa is thought to be derived from this; originally, the kegarë was transferred to the ohnusa, which was then presumably destroyed.
However, the ōnusa is also thought to be derived from the cloth offered to the kami as a gift. In the earliest days of Shinto, cloth seems to have been the standard non-food offering to the kami, and it is still sometimes offered today. That is connected to another way in which haraë is performed. In some cases, kegarë is thought to be transferred to items that are offered to the kami. The kami then removes the kegarë from the item when she accepts it. These days, this is one explanation for the money it is traditional to offer at a jinja when you visit; your kegarë is attached to the money, and then removed by the kami. This, of course, does not fit easily with the idea that kami hate kegarë, but, as I have mentioned before, Shinto does not place a high priority on rigorously consistent theology.
Salt is also used in haraë, often being sprinkled on the people to be purified. Salt is believed, in Shinto as in many other cultures, to have purifying properties, and it is one of the standard offerings to the kami. Sometimes, the salt used in haraë is dissolved in water, which connects it to misogi.
Full misogi involves stripping naked and immersing your body in water, ideally natural water from the sea or a river. This is unusual in contemporary Shinto; One of the few examples I am aware of is at the Okitsumiya of Munkata Taisha in Kyushu, where a whole island is the sacred enclosure of the jinja, and you must strip completely naked and purify yourself in the sea before setting foot on land. (Women are not allowed on this island, Oki-no-shima, at all, which caused some controversy when it was designated a World Heritage Site.) Much more common is for men to wear a white loincloth and women to wear a short white tunic, and then immerse themselves in the water. Although this is much more common, it is still not an everyday practice; such misogi is almost always associated with a particular matsuri. For some unfathomable reason, a lot of shrines have these in the middle of winter.
Misogi and haraë are often both traced back to the same mythological event, when Izanaki returned from Yomi-no-Kuni, the land of the dead, where he had been seeking Izanami, his wife. As death is a source of kegarë in Shinto, he had picked up a lot of kegare there, and he purified himself on the seashore. First, he stripped off and threw away all his clothes, then he immersed himself in the water. Discarding his clothes suggests that the kegarë was transferred to them, thus making it a form of haraë, while the immersion in water is clearly misogi.