The atmosphere serves several purposes for us here on Earth.  For starters, 21% of its composition is oxygen, which all animals must have to conduct cellular respiration, the process that helps cells manufacture energy.  1% of the atmosphere contains, among other gases, carbon dioxide, which is used by plants in the process of photosynthesis, the process which creates energy for all the plant cells.  The atmosphere also has a layer called the stratosphere, which has the ozone layer in it.  The ozone layer screens out harmful radiation from the sun.  If it were not there, we would have to wear radiation suits to go outside.  If Earth did not have an atmosphere, it would affect the water cycle, which is the redistribution of water across the Earth in a cyclical fashion.  Finally, the Earths surface would probably bear a closer resemblance to that of the moon, with all its craters.  The atmosphere causes most meteorites and other space debris to burn up before they strike the Earths surface.