Ethnocentrism and cultural relativism have a few things in common but are quite different attitudes in thinking about culture.
Ethnocentrism, as a term, can be understood by breaking it into two parts. The first, "ethno," implies ethnicity or nationality. "-Centrism" is the idea that something is central to experience or perhaps is the most important factor in a person's perspective. Altogether, ethnocentrism is the attitude that one's own cultural, ethnic, or national experience is the hegemonic experience or the most important. Often, when people act or think in a way that is ethnocentric, they aren't doing so on purpose—a lack of contact with alternate cultural experiences hasn't "opened their eyes," so to speak. On the other hand, people can quite willfully engage in ethnocentrism by asserting that their way of life is better or more important than all others'.
Cultural relativism is a framework popular in philosophy, morality studies, and anthropology. This idea asserts that any particular act, object, feeling, or belief only makes sense in the context of the culture in which it originates. That is, cultural practices must be understood and respected as part of their culture, even if someone disagrees with the practice. A great example of cultural relativism can be drawn from talking about differing food taboos from around the world. Many Westerners are grossed out by the idea of eating insects, snakes, or animals like squirrels or turtles. These are common food items throughout the world, however, so I respect that other people want to eat bugs, even though I don't.
Cultural relativism and ethnocentrism both rely on a cognitive dissonance between "Us" and "Them." It is easy to fool ourselves into thinking that culture, especially our own, is a very fixed phenomena and anything that isn't a part of our culture simply falls into the realm of Other. While this kind of thinking is easy to do, it isn't necessarily true or helpful. Culture exists more like a grab-bag of beliefs and behaviors that evolve throughout time. An Us vs Them mentality is not likely to foster understanding or positive discourse. With ethnocentrism, I think it is quite obviously problematic to consider one's own culture as the only or the best way for things to be. Even cultural relativism is problematic because it does not encourage a discussion of the true value of cultural practices or give attention to the fact that our own feelings about certain beliefs or behaviors, while valid, are also culturally constructed.
In short, cultural relativism and ethnocentrism are similar in their use in the humanities, reliance on an Us vs Them mentality, and the fact that they are quite limiting frameworks. Neither idea encourages open, unbiased discussions about culture.