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.

What is the significance of the title "Twilight"?

Edward speaks on this topic specifically to Bella in middle of the book:

"It's twilight," Edward murmured, looking at the western horizon. . . . His voice was thoughtful, as if his mind were somewhere far away. . . . "It's the safest time of day for us," he said, answering the unspoken question in my eyes.  "The easiest time.  But also the saddest, in a way . . . the end of another day, the return of the night.  Darkness is so predictable, don't you think?"  He smiled wistfully.

So, there you go, the safest time.  In some ways, the most beautiful and the saddest time.  Edward's answer is cryptic, however.  My opinion burgeons from here into the idea that vampires can't be seen in sunlight, . . . and perhaps night is when their greatest lust for blood surfaces.  Therefore, Edward is correct in his assessment.

Also, at the prom, there is a point where Bella indicates that she thought Edward was going to change her that night (she had no idea he was taking her to the prom).  He says to her that her life has barely started, and she shouldn't be ready to give that life up, or for this moment to become the twilight of her life.

I believe it is explained somewhere in the book as the time of day that is most easy for vampires, yet is also the saddest time of day, because it symbolizes an ending. Vampires, as immortal, are stuck somewhere in between life and death. They are forever stuck in "twilight," not in the sun or the daylight of life, yet not in the complete darkness of death. Rather they are stuck between, in the twilight of life, where everything fades to a dim grey.

This question should be on the discussion board because I think the significance of the title is more of an interpretation than a definitive answer.

Having said that, in my opinion, the title has to do with the fact that vampires like the Cullens are able to come out into daylight only on days when the sunlight is no brighter than it is during twilight--the minutes just before sunset and just before daybreak. That is why they choose to live in the northwest, where the skies are overcast most of the time. Unlike in most other vampire stories, the characters in "Twilight" do not burst into flame when exposed to sunlight. Instead, they have another unique feature that would expose them for being different from human beings.

I hope this helps!