The premises are either:
a) logically related to the conclusion
b) irrelevant to the conclusion
When the premises are unrelated to the conclusion, the argument is not valid. This form of fallacy is termed non causa pro causa, or false cause fallacy, the error comprises claiming that something is the cause of an event, despite its not actually having been demonstrated to be the cause. Non causa pro causa takes the forms cum hoc ergo propter hoc or post hoc ergo propter hoc.
In cum hoc ergo propter hoc, the fallacy comprises the assertion that two events that occur together must be causally related (ignoring other possible causal factor/s).
The fallacy of post hoc ergo propter hoc is similar, except that an event is assumed to be the cause of another event merely because it preceded that event (again ignoring other possible causal factor/s).
For the case where premises are logically related to the conclusion, a truth table shows the possible relationships of A (true/false) and B (true/false). Two fallacious variants occur:
Affirmation of the consequent "A implies B, B is true, therefore A is true."
Denial of the Antecedent "A implies B, A is false, therefore B is false."
Many fallacious theistic arguments for the "existence" or "interference" of God rely on these fallacies.
"Stalin's regime committed atrocities after state atheism was instituted. Therefore atheism is bad because it causes evil acts too." [type]
"If I saw a fish turning into a man that would certainly prove that evolution was true. But I have never seen a fish turn into a man, so evolution is not true and we must have arisen through God's creation."[type]
"Society became more sexually promiscous before 2001. Therefore the 9/11 attacks on the twin towers were God's retribution for secularism."[type]
"If an intelligent supernatural being created the universe, then we would see order and organization everywhere. And we do see order, not randomness – so it's clear that the universe must have had a designer." [type]
"I studied hard and prayed to God, and I passed my exam. So God helped me to pass my exam." [type]
Here's an example that combines fallacies:
"If scientists could prove that macroevolution can happen by chance, then that would certainly prove that evolution is true. But I do not believe that scientists can prove this, so evolution is a fiction and God must have created us."