What does the word "Pagan" mean in a modern context? What does it mean being a Pagan?
First, a little history. The word "Pagan" is derived from the Latin word "paganus", which means "country dweller" or "peasant". In the Christianized Roman Empire, in the 4th and 5th centuries, paganus became slang among the military officers meaning "civilian" (that is, a person not enrolled in the army of Christ). Christianity spread much quicker in cities, so that by the Middle Ages, the word for "country dweller" became synonymous with one who was not a Christian.
Today the dictionary definition of Pagan is "a person holding religious beliefs other than those of the main world religions". That's from the Oxford American dictionary. Yes, that describes adherents of Wicca and Asatru for instance, but it's not terribly specific, as it applies to millions of people whose religions don't really fall under the same umbrella as Wiccans and Asatruar.
What we are really talking about is Neo-Paganism. Neo-Paganism is a group of religions that have beginnings in and elements from the 20th century, while building upon ancient roots. Nature worship, mythologies from antiquity, shamanism, occult works from the Middle Ages and Industrial Age, modern environmentalism - all these elements combine to help form Neo-Paganism. Neo-Pagan religions are either polytheistic, panentheistic, or pantheistic. Polytheism is the belief or worship in multiple deities. Panentheism is the view that God is transcendent and immanent (inherent or indwelling) within all creation or that God is the animating force behind the universe. Pantheism is the view that God is all; the universe, nature, and God are equivalent.
Besides these "formal" descriptions, modern Paganism, to me, means paying attention to Nature and recognizing the harmony in natural things. Observing the cycles of the earth, like the sunrise, sunset, the Solstices and Equinoxes, the moon phases. It means having a belief in or adopting a worldview that there is a Force or Forces around us that can help us or guide us if we contact it/them, whether they are called gods or goddesses, elementals, or angels. It means taking responsibility for your words and actions. It means being connected to the "here and now", living "in the moment", while working towards goals that improve your life and your world. Being a modern Pagan means to feel awe and wonder at Nature and life, whether you live on a farm, in a village, in a suburban house, or in a skyscraper in a global city.
For more specific beliefs, values, and practices, you can consult practitioners or texts of particular Neo-Pagan religions, such as Wicca, Asatru, Druidism, Stregheria, etc.