There are two general approaches to belief justification: 1) examining the relationships that exist between beliefs, and 2) examining the way a belief is obtained. These form two competing theories of belief justification -
Internalists all agree that we must look within a belief system for justification, but they disagree about what kind of logical relationship we should look for. Some argue that we should look for a linear chain of belief justification that terminates with beliefs that are self-justified (or self-evident). These self-justifying, or terminal, beliefs are thought to form a foundation from which all other beliefs ultimately derive their justification. This theory of belief justification is called Foundationalism.
A simple example of Foundationalism can be found in Euclidian Geometry. You begin with a set of postulates or axioms that are self-evidently true, then proceed in a linear fashion to derive other beliefs whose truth will be dependent on how they were derived from the foundational beliefs. The foundational beliefs (or axioms) are lending their truth value to all the beliefs that are derived from them. Since the foundational beliefs are obviously true, any belief that is properly derived from them must also be true.
Foundationalism has the virtue of being deductive, so any belief that can be properly derived from the set of foundational beliefs will necessarily be true. However, this entire approach depends on the existence of a set of self-evident beliefs. Some philosophers are dubious of positing a set of beliefs that are free from the requirement of justification. They do not believe there are any beliefs so special that we cannot reasonably ask, “why should I believe that?”
Coherentism is the internalist theory of belief justification that rejects the existence of terminal beliefs in our belief system. Coherentists believe that every belief in our belief system must be justified by its relationship to other beliefs we hold. So instead of a linear model of justification, they suggest a web-of-belief where each belief (or set of beliefs) must be justified by its relationship to the other beliefs we hold. The more interconnected, or coherent, our beliefs are, the stronger the justification. So, like the foundationalist, the coherentist is looking for internal logical relations between beliefs, but they reject the idea that some beliefs have a special or privileged status in the system; every belief must be justified in order to be rational.
Externalism is a theory of belief justification that holds the way beliefs are acquired is more important than the internal relations between beliefs.
Externalists reject Internalism for two reasons: 1) they are dubious of any belief being self-justified, and 2) they are not convinced that the logical coherence of beliefs is sufficient to establish the truth of the beliefs within the system.
As an alternative, externalists look to how we get our beliefs as the most important factor in determining whether or not a belief is true. They believe there are legitimate and illegitimate processes by which beliefs can be formed and we should focus on the legitimate processes.
The externalist approach focuses on the senses we use to perceive the world around us, and the way sensory information is processed in the brain. Thus, externalism focuses on neurophysiology as the key to acquiring true beliefs. That is, beliefs will be justified if our senses and brains are working correctly. When are senses are impaired, either by poor observational conditions, or when our brains are impaired by some set of internal abnormalities (e.g., drunkenness, insanity, sleep depravation, etc.,), we are likely to form beliefs that are not reliable. So, for the externalist it is the process that is more important than the internal logical status of the beliefs already in our minds.