Physics is characterized by a handful of "fundamental constants." For instance, the gravitational constant, G, tells us the force of gravity between different masses. Similarly, the "fine structure constant" gives the electric force between two charges. Does our existence depend on the values of these constants? If they were different, could life still exist?
These kind of speculations fall under the heading of the "anthropic principle," the idea that somehow the values of physical constants must have their current values, or we would not exist. This argument occurs in both a scientific and religious context. Some physicists invoke the anthropic principle to "predict" the values of the constants of nature, arguing that if these constants were very different from their measured values, life itself would be impossible. (The astute reader will note that this is a postdiction, not a prediction). Similarly, some religious thinkers have argued that the universe is "fine-tuned" for the existence of life.
While I am, in fact, Catholic, I do not find either the scientific or religious anthropic argument to be particularly compelling. (I guess I should be called a misanthrope). These kind of arguments seem to me to be a form of ex-post-facto reasoning. It's particularly telling that scientists often invoke the anthropic principle only after they have run out of all other explanations. But my main objection to these arguments is that they display a lack of imagination. After a long immersion in science fiction, it's easy to believe that we have no idea what sort of life might be possible, and it's rather limiting to assume that our form of life (which might, indeed, depend on the universe being "just so") is the only possible form of life.
Fictional consequences of parallel universes with different fundamental constants are explored in Isaac Asimov's very famous novel, The Gods Themselves (which is the earliest SF treatment of this subject that I know of) and later in Stephen Baxter's Raft. My present post was inspired by a paper that appeared on the arXiv today by Fred Adams, called "Constraints on Alternate Universes: Stars and habitable planets with different fundamental constants," which you can read here. Fred explores what the universe would look like with vastly different values for the gravitational constant and the fine structure constant, and suggests that life as we know it would be possible even if these constants are varied by several orders of magnitude. But gravity always has to be much weaker than the electric force in order for stars and planets to form.
A related topic is whether or not the constants of nature could change with time. There is a huge scientific literature on this subject, including a few papers I've written myself. I suggest you take a look at my paper on the possible time variation of pi, which appeared on Apr. 1, 2009.