A black hole is a region of space in which the gravitational forces are so great that even light (and other forms of electromagnetic radiation, except in certain special cases) cannot escape its pull. The existence of black holes were predicted theoretically from Einstein's theory of relativity, first realized by geologist John Michell in 1783, before any observations were made confirming their existence.
Black holes are singularities (points of infinite density and zero radius). Currently, our sciences break down at the singularity, even if relativity makes a good attempt. The singularity is an unknown and unresolved phenomenon in reality because of this.
Similar to the Big Bang
Like black holes, the initial object that preceded the Big Bang was also a singularity, of a different sort. Knowing that, we also know that our current scientific understanding of how the universe works breaks down there, as well. This has dire implications for the various cosmological Arguments, since they tend to attempt to project common understandings of how the universe works into areas where those understandings knowingly fail.
Finely tuned for Black Holes
Black holes also have an impact on the fine-tuning arguments, where the argument is that the universe is finely tuned for we humans to exist. If anything, the universe appears to be finely tuned for the production of black holes - the most dangerous and destructive objects in the universe. Outside of Earth, most of the universe is harsh and inhospitable like this.
Example of the Scientific Method
The vast majority of the arguments for the existence of god are examples of trying to logic God into existence, where, absent of any empirical evidence for any kind of confirmation, the apologists attempt to "prove" their god. As noted in the introduction, the existence of black holes was considered merely hypothetical, despite compelling logical inductions from already known physics that suggested they might exist. It wasn't until we gathered several kinds of empirical evidence directly supporting the existence of black holes that the scientific community accepted their existence.
In contrast, those apologist arguments make educated guesses, but ultimately do not confirm the assertions with empirical evidence, and declare them absolutely proved, anyway.