I would like to offer some background to those who may not have a good idea of Paganism in general. Please keep in mind that there is NO such thing as what Pagans believe, Paganism is an incredibly diverse grouping of religious thought. The only thing you really can do is point to commonalities and differences in order to gain an understanding of what drives the Pagan heart.
The word Pagan derives from the Latin word paganus which means literally, country dweller. In the early history of the Christian Church during the time of Constantine, the legislated conversion of people to the new religion took place first in the cities where it was easier to enforce. The country folk retained their chosen faith much longer, thus the word for country folk became synonymous with non-Christian.
The Generally accepted Tree of Modern Paganism includes many religions. These religions are generally differentiated as "Old Pagan" and "Neo-Pagan".
The "Old Pagan" branch is those religions for whom there ins an unbroken tradition stretching back into history. Examples of these would be Native NAmerican Aboriginal, Hinduism, and African Shamanism. The "Neo-Pagan" branch is those religions recently created, or reconstructed after having been lost. Examples of these would be Gardnerian Wicca, Druidism, Odinism, and Celtic Reconstructionism.
There are three relatively common philosophies which tend to define Pagan religious thought. Most Pagan faiths contain some measure of these three ideas, although some may emphasise one over the others.
1) Pantheism
pantheism
\Pan"the*ism\, n. [Pan- + theism.] The doctrine that the universe, taken or conceived of as a whole, is God; the doctrine that there is no God but the combined force and laws which are manifested in the existing universe; cosmotheism.
Source: Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc
Pantheism is the idea that the Divine within all things, that each object, creature, and person together contains a portion of that which altogether makes up the divine. This is one of the factors which contributes to the idea of earth-based religions. Since the Divine is in every object and creature, since the Earth herself is divine, it behooves us to take special care of the Earth.
2) Polytheism
polytheism
\Pol"y*the*ism\, n. [Poly- + Gr. ? cf. F. polyth['e]isme.] The doctrine of, or belief in, a plurality of gods
Source: Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc
Polytheism is the idea that more than one god exists. Most Pagan faiths believe in at least two distinct deities, God and Goddess. Some like Celtic and Hindu, believe in hundreds. Almost all believe that Gods and Goddesses outside their own pantheon also exist, either like the Romans, who believe most Gods and Goddesses of other religions were just renamed Roman Gods and Goddesses, or like ancient Europeans who believed different Gods and Goddesses held sway over different parts of the earth. Polytheism is one of the factors which allows modern Pagans of different faiths to worship together in joy and harmony. Our Gods and Goddesses are not jealous.
3) Animism
animism
\An"i*mism\, n. [Cf. F. animisme, fr. L. anima soul. See Animate.]
2. The belief that inanimate objects and the phenomena of nature are endowed with personal life or a living soul; also, in an extended sense, the belief in the existence of soul or spirit apart from matter.
Source: Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc
Animism is closely related to Pantheism. Animism teaches that each object, creature, and person is endowed with a living spirit. This combined with Pantheism results in a universe in which each object is both a part of the divine and an entity in its own right.
These three principles are commonly called the Three Pillars of Paganism. They are not universal but the vast majority of Pagan Religions contain all three to a greater or lesser degree.
Another very common principle is the idea that all paths lead to the Divine. This is not so much a nod to relativism as it is an acknowledgement that the nature of the Divine is not completely knowable by humans, that all religions are simply human interpretation of a concept to large for the human mind to grasp directly. Therefore no religion is the absolute truth but rather each is a way of understanding the nature of the Divine in ways our minds can grasp. Since each religion is an interpretation, no one religion is the truth, nor is any religion false. This leads to a cooperative nature among some very radically different belief systems that would be impossible among the Abramic religions, even among the practically identical Christian sects. Modern Pagans of vastly different belief systems commonly gather to worship and fellowship, emphasising their common principles rather than their differences.