Reading : Western Philosophy V.2 "The Five Proofs of God"
Aquinas puts forward five arguments to prove that 'God' exists. Each ends with a phrase along the lines "and this we call God". Aquinas is not trying to argue for the God of a specific religion, and so none of his arguments is an attempt to support scriptural teaching (apart from that which is common to the view of any theist). His arguments are for a "prime mover" or "prime cause" which he says must have existed for anything else to exist.
The first three reasons can perhaps be summed up as "something cannot have come from nothing" - i.e. we can tell that the universe exists and is active, and since everything in our experience can be seen to have been caused by something else, then something must have started the universe off "and this we call God". So we have:
For these three arguments, Aquinas says something must have kicked the whole thing off, since it's not possible that a sequence of causes could continue back ad infinitum. However, he does not say why this is impossible, and so although one might be inclined to be sympathetic to this view, it's not an undeniable fact: it is possible to conceive of a universe which extends back infinitely, with never any cause. Current cosmological theory says that the universe started with a "big bang", but there is still room for doubt about whether what preceded that was a "big crunch".
The fourth argument, "degrees of perfection" says that since we grade things in relative terms, e.g. "hot", "hotter", there must be "something which is truest and best and noblest" that we are measuring things against, "and this we call God". This has echoes of Plato's realm of forms, but is dubious: just because you have "hot" and "hotter" doesn't mean there's a "hottest". In fact, the example of heat (which Aquinas uses) is a poor one, since you could never achieve a temperature that could not be increased (at least theoretically).
Fifthly, Aquinas puts the teleological argument: things seem to be guided to a particular purpose, or design, and so there must have been a designer. It seems likely that he has in mind things such as a polar bear's coat being white. This point of view is challenged by evolutionary science, which argues for "natural selection" and Dawkins' "Blind Watchmaker".