All four titles relate to the moon and the changing parts of the day. Twilight is the time of day right before sunset, when the last light of day gives way to night. Edward tells Bella that twilight is the hard part of the day for him, because it's the time when darkness comes, and it reminds him of his lifetime of endless night/darkness as a vampire. Later in the book, he also talks about how Bella wants that to be the "twilight of her life," because she wants him to end her human life and turn her into a vampire. But he isn't ready to end her life when it has barely just started.
The new moon is when the moon goes dark; it's the time of the lunar cycle when the moon is closest to the sun. In this book, the "new moon" refers to the darkness Bella feels when Edward leaves, as well as the transformation of Jacob Black into a wolf. The link between wolves and moons is fairly common, and since this is a new life for Jacob, it links well to the title.
Eclipse is what happens when the sun and moon line up together. One eclipses the other, blocking it out from sight to the people on earth. There is a quote about this in that book, but I can't find it right now, unfortunately. In general, you can think of the idea that in this book Bella feels very torn between Jacob (her sun) and Edward (her moon). One will have to "eclipse" the other for her eventually, when she decides between them.
These are all just my own interpretations; I'm sure other people's ideas will vary.