The argument begins with the existence of the cosmos and argues that its existence requires that there be a first cause. Thus, the argument is a causal argument, nevertheless it is called the cosmological argument. Thomas Aquinas (1225-1274) has stated the argument in its most famous form. His philosophy, called Thomism, was initially resisted by the Catholic church. Later it became the "official" philosophy of the Roman Catholic Church and remains very influential.
He gives five arguments. The first attempts to show that there must be a First Mover.
The second, which is the most popular, attempts to prove that there must be a First Cause.
"The second is from the nature of efficient cause. In the world of sensible things we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or one only. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate, cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God."
(Note: Although this is a causal argument and draws on our intuition that things must have a cause, it is nevertheless called the cosmological argument since it tries to account for the cause of the cosmos.)
The third way is important, since it tries to show not only that God exists, but that he is a NECESSARY BEING.
"We find in nature things that are possible to be and not to be [i.e., there are contingent beings].....But it is impossible for these always to exist for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already in existence. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence - which is absurd. Therefore, not all beings are merely possible [contingent], but there must be something the existence of which is necessary....This all men speak of as God."
The fourth way is that there is a Perfect Goodness, and the fifth that there is an Intelligent Governor of the World. The arguments are from Summa Theologica,Part I.