The most common understanding of “justification” within an epistemological context is that one can be justified to have a belief if they have sufficient reason for believing it. The act of justification is the process of figuring out what makes the most sense to believe given all available evidence. An idea is justified if one can reasonably conclude that it is true after considering all evidence that is available.
The word “justification” by itself has the potential to confuse some readers because this word is closely related to the words “justice” and “just”, each of which seem inseparable from normative (essentially value-driven) concepts such as obligation, rights, and responsibility. Because of this, I often choose to use the term “epistemic justification” or “epistemically justified” or something similar to this as opposed to just saying “justification” or “justified”.
The most important question is what factors determine whether a belief is epistemically justified. There are different factors that determine justification for different forms of knowledge. The brain’s innate knowledge, also known as synthetic a priori, is always justified even though such areas of knowledge are less of cognitive processes as cognitive facilities (knowledge that is innate to the brain is not the product of brain functions). Also analytic a priori knowledge (deductive inferences, etc.) is justified because it is logically implied by synthetic a priori knowledge.
For perceptual knowledge, justification is based on coming to the most reasonable conclusion regarding what one is perceiving. We can also be justified in our beliefs regarding the laws of nature by first making careful and repeated observations and then using inductive reasoning. Claims made by others can be justified if a number of factors regarding the nature of the claim and the claimant are met.
Worldviews differ in the degree to which they are epistemically justifiable versus the degree to which they rely more on faith. For example, atheism is more epistemically justifiable than Judaism, Christianity, or Islam because it does not rely on scriptures, which the latter three worldviews all have as integral parts and are to be taken, at least to some extent, as the word of God on the grounds of faith.