Don't worry, it probably won't -- at least, not for another 5 billion years or so. But if it did, a very interesting thing would happen: our atmosphere would freeze, and precipitate out into oxygen and nitrogen snow. (That's an interesting image: our atmosphere snowing onto us.) In fact, it would look a lot like normal snow, since solid oxygen is clear with a very pale sky-blue color, and solid nitrogen is clear and colorless. Difference is, there would be a lot more of it than we're used to. For the math stuff, we turn to ask an astronomer:
"How much snow would there be? The earth's atmosphere has a mass of about 5000 trillion metric tons (this can be estimated using atmospheric pressure, newton's law Force = Pressure / Area = Mass x Acceleration, and the earth's gravitational acceleration and the surface area). If we assume that the "snow" formed when the atmosphere freezes has a density comparable to freshly fallen water snow (about 100 kg/m^3), then we can find the depth of the frozen atmosphere: Depth of "Snow" = Atmosphere's Mass / Density of Snow / Surface Area."
In plain English, this works out to about 100 meters of snow -- more than enough to cover all but the tallest buildings. If our sun went out, (hypothetical) alien visitors who arrived a year or so after the fact would find nothing but an atmosphere-less world covered with icy snow, a blank slate save for the occasional Sears Tower poking out of the tundra. Creeeeepy.