The Cosmological Argument attempts to prove that God exists by showing that there cannot be an infinite number of regressions of causes to things that exist.  It states that there must be a final uncaused-cause of all things.  This uncaused-cause is asserted to be God.
     The Cosmological Argument takes several forms but is basically represented below.

Cosmological Argument

1. Things exist.
2. It is possible for those things to not exist.
3. Whatever has the possibility of non existence, yet exists, has been caused to exist.
   1. Something cannot bring itself into existence since it must exist to bring itself into existence which is illogical.
4. There cannot be an infinite number of causes to bring something into existence.
   1. Because an infinite regression of causes ultimately has no initial cause which means there is no cause of existence.
   2. Since the universe exists, it must have a cause.
5. Therefore, there must be an uncaused cause of all things.
6. The uncaused cause must be God.

     Thomas Aquinas (1224-1274) had a version of the Cosmological Argument called the Argument from Motion.  He stated that things in motion could not have brought themselves into motion but must be caused to move.  There cannot be an infinite regression of movers.  Therefore, there must be an Unmoved Mover.  This Unmoved Mover is God.

Strengths of the argument

     The strengths of the Cosmological Argument lie in both its simplicity and easily comprehensible concept that there cannot be an infinite number of causes to an event.  Some arguments for God's existence require more thought and training in terms and concepts, but this argument is basic and simple.  Also, it is perfectly logical to assert that objects do not bring themselves into existence and must, therefore, have causes.

Weaknesses of the argument

     One of the weaknesses of the argument is that if all things need a cause to exist, then God Himself must also, by definition, need a cause to exist.  But this only pushes causation back and implies that there must be an infinite number of causes which cannot be.  This is paradoxical.  
     Also, by definition, God is uncaused.  

Courtesy of http://www.carm.org/apologetics/cosmological.htm