It depends on the order or congregation really. Usually a woman will contact the person in charge of vocations for whatever community she wishes to join. Some visits are made, conversations are had, and an application is made. I'm not sure exactly what the application would be like, as I've never applied to be a nun, but I imagine it would not be too different from my seminary application. They'd ask what sort of life the woman has lead, what her prayer is like, why she intends to become a nun, and other questions like that. If she is accepted, she will go through a similar process to what most aspirants to the religious life undergo. First she will undergo a period known as postulancy. The length of this time varies, but I think that a year is probably not an unusual amount of time, but I may be wrong here. A postulant makes no vows, perpetual or temporary. It's a time to examine whether or not the religious life really is for you, and whether you intend to go forward with this. If a woman discerns during her postulancy that she should stay, she will go on to the novitiate. Here she will make temporary vows of chastity, poverty, and obedience. As a novice she will live as a nun in the community, and thus become acclimated to the life and grow spiritually. After some time (6 years is usual, I think) she will make her final vows and become a nun.
The woman will first go through a period of initial contact with the community she wishes to join. This is often about a year, could be more if there are circumstances preventing sooner entrance.
At entrance, the woman becomes a postulant. She still wears normal clothes and the purpose of the postulancy is to gradually experience the life of the community and discerning with the community if it is the place for her - this discernment is both on the part of the woman and the community itself. It can last between anywhere from six to eighteen months, depending on the community in question and that time is usually flexible.
After completing postulancy, the woman will enter the novitiate. This is a more formal step but she will still not take vows. This is when she will first receive the habit and veil, usually white as opposed to the black veil of a fully professed but it can differ by order. In communities where they take religious names, this would happen at this point as well. The novitiate is mostly a period of study and lasts generally about two years.
At the end of the novitiate, temporary vows are taken. Until this point, the woman is free to leave at any point but with temporary vows they are bound to the community for three years. They take on more responsibility in the community at this time.
After three years of temporary vows, the woman may now make the solemn vows which bind her to the community for the rest of her life.
There are slight variations by order and community but generally it will follow these steps.