A society is a coherent group of people with similar lifeways, but it's a larger conglomeration than a culture. For me, trained as an anthropologist, the distinction is one both of scale (a culture can be pretty small) and difference (many cultures can make up a society). Usually, a society contains more than one class of people -- like urban and rural, rulers and ruled, clerics and laity...
Civilization usually refers to a society (not merely a culture, see below) that is located in an urban administrative and/or military center. While many sociologists use the term civilization to denote a particular level of cultural production, the more generic meaning is city-based society. With a city, the distinctions I mentioned above are always present to one degree or another. Civilization means division of labor (urban manufacturing versus agricultural production with a strong separation of those tasks usually based on hereditary distinction; distinct modes of urban manufacturing with strict boundaries about who can learn what; a leisure class, like an administrators, military officers, and/or priests) and hierarchy. Civilization means monument building (irrigation systems, temples/tombs, walls), which also means a certain amount of conscripted labor.
All civilizations are societies, but not all societies are civilized.
