The circle may be the one thing that is universal among neopagan traditions. That does not mean, though, that the ritual circle bears the same meanings for all of them.
However it is cast, it is called the circle even though most pagans recognize that a circle is a two-dimensional figure, while they are in effect casting a three-dimensional (at least) structure to define and protect the space they will be working in. The language is inaccurate, as in truth they are casting a sphere (or half sphere) of three or more dimensions. Nevertheless, it is called a circle.
The purpose of the circle is to define and protect the space that is set aside for ritual, celebration or magical work. Typically, the circle, or the space it defines, is said to lie outside of normal space and time, which are often viewed as being suspended for the duration of the proceedings within it. Many traditions urge their members to visualize the circle as a shimmering, translucent wall or dome of electric or sky-blue energy. Some even claim to be able to see it this way.
With this generic definition, then, the circle is a boundary that transports those within it to a timeless, spaceless "place between the worlds" where divination and magic become possible, or more accessible. In addition to defining this magical, sacred place, the circle also functions as both a wall of protection from evil and/or undesired energies, beings and influences, as well as a portal for whatever is desired or needed to come into the sacred space.
Circle etiquette varies from tradition to tradition. Some have the high priest and/or priestess cast a nearly complete circle while everyone else waits outside. Then they are brought in and the circle casting is finished. Some cast a complete circle and then cut an entrance or open a door in the circle to let the rest of the participants inside. Most probably assemble everyone in the ritual space and then cast the circle with everyone already inside it. Once cast, leaving the circle is discouraged, if not forbidden. The groups that do not forbid it have set procedures for cutting a door (and closing it behind) if someone needs to leave, to go to the bathroom, for instance. This procedure is reversed to re-enter the circle. Some groups avoid the whole issue by casting, or extending, the circle to include an entire house or area that contains any place participants may need to use.
Those attributes of the circle are common to almost all modern pagan traditions. But there are more.
One of the reasons the circle has special meaning is that, as a geometric figure it has no beginning or end. Thus, while it contains a finite area, the circle itself is infinite. The circle is the form of zero, our mathematical symbol of nothingness. But nothingness is very special because it is the mother of everything. In sacred geometry, all numbers–which represent all things–are constructed in turn within, or by making use of the geometric properties of a circle.
The center-point of the circle represents the number one. One arises spontaneously simply because every circle has a center. For this reason, it is said that zero (non-beingness) and one (beingness) lie within every other number. If you measure the radius of the circle (the distance from the center to the perimeter), and then draw another circle with the same radius using a point on the perimeter of the original circle as the center-point for the second one, you get two intersecting circles. The common area that both circles share is called a vesica piscis. Assuming you have drawn your circles next to each other instead of one on top of the other, the center-points of the two circles can be connected by a horizontal line. This is the number two. The vesica piscis is shaped somewhat like a womb, and that is exactly what it represents in sacred geometry because it is vital to the construction of the rest of the numbers. If you draw connecting lines from each center-point, at each end of the horizontal line, to the apex of the vesica piscis, you get an equilateral triangle. Three. Three is the first number to emerge from the womb that was created by generating the number two. So three is the first-born of the parent numbers one and two. Three is also the first number that can manifest in the material world, since material things have three dimensions.
An excellent source for learning how to construct the rest of the numbers, and their qualities in sacred geometry as well as in nature is A Beginner's Guide to Constructing the Universe, by Michael S. Schneider.
The circle represents infinity as well as the source of all of existence. It is literally the womb of the universe. By association it also represents any womb. Being the source of everything, the circle is the power of generation or manifestation. It is the power and mystery of creation. But, in the material world at least, all things eventually decline and return to the source. So the circle is also destruction. The powers of creation and destruction, and of life and death, are symbolized figuratively in Wiccan and related traditions by the cauldron of the Hag or Crone.
The death aspects of the circle connect it directly to the grave (the womb of Mother Earth) and the underworld.
One of the basic principles of neopagan traditions is that nothing is ever truly lost. Energy cannot be created or destroyed, and everything is made of energy. In the natural world all material things decline and eventually die, but this is really only a shift in the form of the energy that comprised the matter. As the physical form dies its energy returns to the source, the cauldron, where it is reconstituted and recycled. Then it emerges into the world again in another form or body.
So the circle also represents regeneration, transformation, transmutation and reincarnation.
Consider the sky. From our vantage point on the surface of the Earth, the sun, moon and stars seem to revolve around us in a giant circle or sphere. Closer observation reveals patterns. The moon waxes, comes full and wanes, repeating this cycle every twenty-eight or twenty-nine nights. The sun (and other bodies, too, but it is most obvious with the sun) moves laterally on the horizon, rising each day slightly to the north or south of where it rose on the previous morning. Except for two special days, when it reaches its maximum northern or southern point. The movement of the sun in this manner corresponds to the cycle of the seasons. The extremities of the sun's journey along the horizon mark the summer and winter solstices, while the mid-point between them marks the spring and fall equinoxes.
These key points and the dates they correspond with can be plotted on the perimeter of a circle, and by drawing connecting lines along the axes of the solstices and equinoxes the year is neatly divided into four quarters that correspond to the seasons. It is simple to see how this kind of representation gives birth to the concept of time as a cyclic phenomenon. Everything (well, most things) moves in a predictable cyclic pattern. And although each individual life seems to move in a linear fashion from birth, through growth to maturity, then through decline and into death, observation of the patterns of life show that this is simply an illusion. Life proceeds in cycles too. Throw this in with the belief that nothing is ever truly lost, but only recycled through the source (the cauldron) to be reborn again, and you have the makings of a systematic spirituality whose primary symbol is the circle because the circle symbolizes everything–being/non-being, creation/destruction, life/death, time, seasons, rebirth, even history.
Before long myths are created to explain all of these cycles and correspondences, and to teach about the seasons (winter comes when Demeter goes into mourning for Persephone, and spring arrives when Persephone returns from her time in the underworld) or to transmit specific knowledge about the sky (Scorpio and Orion lie opposite each other in the sky because when the great hunter boasted that nothing could kill him the scorpion stung him and he died.).
Work at this long enough, and you eventually get a Wheel of the Year, a circular mandala that illustrates the cycles of the seasons and acts as a repository for the myths that preserve the knowledge base of a culture.