One of the most common discussions on this site is, unsurprisingly, about the possible existence of any gods. "Does God Exist?" is a common topic not just around here, but all over. Most assert that some god definitely exists, a few question that claim, and fewer still aggressively assert the opposite.
In all such discussions, you will find the same basic arguments repeated over and over. There is a limited number of arguments offered for the existence of gods, although there are certainly variations and ways to make old arguments look new. For that reason, it is a good idea to familiarize yourself with the most common this will make it easier to deal with them when you encounter them.
Most purport to prove that a single, personal, creator-god exists, so that is primarily the sort of argument which will be discussed here, but, a few are compatible with other sorts of gods.
The Cosmological Argument covers a lot of ground and takes a number of different forms, although the most common deal with two ideas: that the existence of the universe requires God as an explanation (First Cause - this form is also called the Etiological Argument) or that order in the universe requires God as an explanation.
The Argument to Design is referred to as the Argument from Design because the person is arguing from the existence of "design" in the universe and to the logical conclusion that the design requires a "designer" God.
The Ontological Argument is one of the oldest arguments in Christian theology attempting to prove God exists. It is also one of the most difficult to understand because it relies purely upon logical considerations and not upon empirical evidence.
The Arguments from Morals and Values are two separate, but connected arguments for the existence of God. Together they make up what are known as the Axiological Arguments (axios = value). They assert that the existence of values and/or of morals proves that God exists.
The Argument from Religious Experience starts from the premise that all our knowledge of the world essentially relies upon experience. As a result, religious experiences should be accorded the same basic validity as other sorts of experiences. Thus, people claiming to have experienced god, angels, miracles, etc. should be believed. Based upon this evidence, the existence of a god is made, at the very least, quite likely.
The Argument from Common Consent argues that belief in some sort of god is innate or instinctive and has existed consciously in nearly the whole of humanity throughout history. The best way to explain this, or so the argument goes, is to assume that some sort of god really does exist after all. Belief in a god wouldn't be so popular or pervasive if some god didn't exist, therefore some god must exist.
According to the Argument from Reward, people who believe in God are happier than those who do not. This implies the extra happiness is a reward from God and, hence, God must exist.
The Argument from Justice starts from the premise that in this world virtuous people are not always happy and do not always get what they deserve while wicked people do not always get the punishments they should. The balance of justice must be achieved somewhere and at some time and that since this does not occur here, then it must occur after we die.
According to the Argument from Consciousness, neither naturalism nor materialism can give an adequate explanation of mental events like consciousness. Consequently, divine and supernatural explanations are needed to explain why we are conscious and how our brains work.
The Argument from Miracles is based first and foremost on the premise that there exist events which must be explained by supernatural causes in short, some sort of god. Probably every religion has had miracle claims and so the promotion and apologetics for every religion has included references to allegedly miraculous events. Because it is likely that a god is their supernatural cause, belief in this god is supposed to be reasonable.
What is the Argument from Scripture? Sometimes, you can find popular apologists trying to argue that their particular holy books qualify as evidence supporting their claims that their god(s) exist. One method might be to point to the supernatural events recorded in these books as requiring the existence of a god. Or they might point to how influential the scriptures have been, arguing that this would not have been possible without the help of a god.
Pascal's Wager essentially argues that to believe in God is the best bet because if God exists, you'll go to heaven and avoid hell. If you don't believe in God, you might lose all this. If God does not exist, you'll have nothing to lose. So it's better to believe in God than not to.