As large section of the philosophical branch of epistemology is the question of epistemic justification. Justification attempts to determine what grounds there are (or what justification there is) for one to hold to propositions and beliefs. In other words, justification specifically asks the question as to “How sure do we need to be that our beliefs correspond to the actual world?” So, typically epistemologists wish to determine the reasons behind why someone holds to a particular belief, why a particular belief a true one, and how one knows what one knows. For some, epistemic justification is a motivation to do philosophy. In an informative work I am reading on Bertrand Russell, Russell explains that he took interest in philosophy on the grounds that one could doubt nearly everything, including sense perception. Thus, Russell hoped that by engaging in philosophical thought he would be led to certainty for at least some beliefs (to what conclusion Russell reasoned to I am yet to discover).
Nonetheless, epistemologists will argue that for a belief to be justified, it must have something that justifies it. Philosophers refer to this as a “justifier.” A justifier in this case could be a piece of evidence. If a detective discovers a bloody knife lying on a carpet just a few feet away from the corpse of an apparent victim, he would consider it to be an important item of evidence. Thus, in this scenario the bloodied knife next to the victim’s body is the justifier for the detective’s belief that the man was murdered with the knife. It is also uncontroversial to observe that justifiers come in different strengths and types. Some evidence points more strongly towards a particular conclusion whereas other evidence might not be as persuasive or needs supplementation by yet more evidence. Either way, for a belief to justified, it must have a justifier.
According to a number of epistemologists it should be our intellectual responsibility and obligation to believe what is true and thus avoid believing what is false. Thus, we should emphasize a critical examination and evaluation of our beliefs in order to determine what we are “within our rights” to believe in. And this is where philosophers have disagreement as we can see reflected in the several epistemic theories they have proposed. These theories are as follows:
Foundationalism – This approach says that beliefs can be justified on the basis of basic, foundational beliefs. It holds to two types of beliefs: basic and non-basic. A basic belief is one that is justified without reference to other beliefs. A non-basic belief is one that depends on a basic belief in order to be justified. Basic beliefs require three components to be considered basic:  Evident to the senses,  Incorrigible, and  Self-evident.
Coherentism – This approach implies that for a belief to be justified it must belong to a coherent system of beliefs. In other words, the beliefs that constitute the system must “cohere” with one another. It is opposed to foundationalism in the sense that it asserts that knowledge does not require a secure foundation but is rather established by the interlinking strength of its components. This includes three components:  Logical consistency,  Explanatory relations, and  Various inductive (non-explanatory) relations.
Evidentialism – This approach says that a belief in a proposition is only justified if evidence supports believing in the proposition. The debate within evidentialism hinges on the nature of evidence, what constitutes evidence, and how evidence relates to believing a proposition.
Externalism – This approach says that justification of belief depends on additional factors that are external to a person. An external factor refers to those factors outside the psychological states of an the individual who is gaining knowledge.
Internalism – This approach says that justification of belief is solely determined by factors that are internal to an individual such as the individual’s consciousness.
Epistemological Infinitism – This approach says that beliefs are justified by infinite chains of reasons.
Foundherentism – This approach attempts to justify belief through the combination of foundationalism and coherentism.
Reformed Epistemology – This approach examines the justification of religious beliefs. For example, it argues that belief in God may be considered properly basic.
Epistemological Skepticism – This approach questions the possibility of justified knowledge, and holds that attaining knowledge of anything is difficult if not impossible.
These epistemic theories all have their own ways of explaining how a belief is justified and how we can know what we know. Each theory also has value and raises thought provoking questions that call for further engagement between philosophers.