The term ritual alarms a lot of people, but in the Pagan context it merely denotes a religious ceremony like any other. The problem is that the word ritual too frequently gets coupled with Satanic. Neo-Pagan practices have nothing to do with Satanism (which originated as a deliberate perversion of Christianity).
We will try to answer the most obvious questions. (For more extensive information, we suggest you consult our list of recommended books.)
Neo-Pagans get together to celebrate the changing of the seasons (these celebrations are the witches sabbats), to do magic (a kind of active prayer see magic) on the full or new moon or just to worship the gods.
A typical Wiccan ceremony goes like this: As Wiccans dont use church buildings, they create a temporary temple on the ground, or in someones living-room, or wherever they are. They mark out the area by drawing an invisible circle with a sword or ceremonial knife (or wand or finger) and consecrate the space with the four elements air (usually represented by incense), fire (with a candle), water and earth (usually salt). Then they address the four directions, calling in the qualities of same intellect, will, love and manifestation. And then they call the deities to be present. In Wicca the deities may be a generic Goddess and God or specific forms, usually Celtic or Greek. Next the priest and priestess bless the wine in a mini-ceremony that reminds the participants of the ultimate unity of everything. Bits of business for the season are performed a blessing of seeds in the springtime, dancing around a maypole on May Day, remembering our ancestors at Halloween, or decorating a tree for Yule. Then the participants "raise power" which involves singing and dancing in a circle and then sending off the energy to a specific goal or to the gods. Afterwards they share wine and bread, and then say farewell to the deities and the directions. Other kinds of Pagans hold rituals that are somewhat similar.
The ritual knife (called an athamé pronounced ath-ah-may) is never used for cutting anything material. It may look alarming because it's usually in the form of a dagger, but it represents human will, which can be dangerous (or, in some Wiccan systems, it represents mind, which can also be dangerous). But the careful application of both will and mind are essential to our lives. Also the knife is an essential tool to life from the kitchen and dining table to its myriad transformations, as plow, shovel, needle, axe, scalpel, ad infinitum.
Modern science recognizes many elements, and breaks those down into yet more elemental particles, and beyond, to the mysteries of quantum physics. Still, we humans tend to have old-fashioned minds and the four elements make sense to us. All things, particularly living things, are composed of, and depend on, some combination of air, energy, water and solid matter. We remind ourselves of this as we bless the circle, and then we call the directions, with which we associate the elements and corresponding aspects of ourselves.
This is a large topic. Briefly, you can think of the gods at their most basic as being Mother Earth and Father Sun, whose interaction produced life and the world that we live in. Beyond that, you can think of them as archetypes within the human psyche, and/or energy constellations in the outer world, and/or anthropomorphic representations of the divine (prime mover and/or underlying reality and/or the sum total of All That Is). We feel that many aspects of the divine can be more readily thought of, anthropomorphically, as female than as male and if were going to make the gods in our own image (which, due to the limitations of our minds, we humans tend to do), then lets not leave out half of ourselves.
It varies, but usually involves the priest dipping his knife into the cup of wine the priestess holds (in some traditions the priest holds the cup and the priestess the knife) and they say a few words to remind participants that life comes from the marriage of earth and sun-energy, and that new people are born from the union of men and women and that we create art, science and technology from new combinations of things and ideas.
Robes represent our coming to the ritual as our essential or divine selves as contrasted with the roles we play in everyday life. Not all groups wear special ritual clothes. And a very few groups hold ritual in the nude (called skyclad), thus communing with the gods the way they made us.
We commune with the gods. We get in touch with the unseen aspects of reality. We get joy, blessings and a sense of close community.