The term 'a priori' (Latin, "from what is prior, before", "from the beginning"), when used in reference to knowledge, is, primarily, the idea that awareness, (or consciousness, or sentience), as such, includes a knowledge of things which transcend contingent experience, or more generally in reference to alleged self-evident "facts", "propositions" or "truth" already implicitly assumed to be true without need for verification ("everyone knows, a priori, that this is so"—"assuming a priori that this is so, we conclude...").
|“|| We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
— Declaration of Independence
|“|| My teaching is not mine, but his who sent me; if any man's will is to do his will, he shall know whether the teaching is from God or whether I am speaking on my own authority.
—Jesus Christ, The Bible (John 7:16-17 RSV)
A secondary usage of the term 'a priori knowledge' is the idea that awareness, as such, includes a knowledge of some contingencies of denying that which is known a priori. For example, if 2 plus 2 equal 4 either is, or is modeled after, a type of a priori knowledge, then this knowledge would include the knowledge that 2 plus 2 does not equal 5.
A priori knowledge thus is contrasted with the 'subjective experiences' which obtain as contingencies of merely physically sensible things. For three examples, no one knows, by virtue of being aware in general, that there are two plush dog toys on my desk, that there are stars in the space beyond Earth, and that the air at 40,000 ft above Earth is so sparse that you would soon die of oxygen depletion were you to try to simply breath it. Those propositions are part of a posteriori knowledge.
The cognitive nature, and central problems with, a priori knowledge
For limited sentient agents, abstraction is the neurological function of allocating the least possible explicit knowledge of, and, hence, the least neurological energy to, the correspondence between the abstraction and that from which the abstraction is abstracted. In other words, for the sentient creature, thoughts and ideas, including the process of forming ideas from percepts, is a matter of efficiency in the use of limited active cognitive resources.
But, in so far as a given depth of a priori knowledge requires little, if any, cognitive energy to apply either to practical concepts or to empirical pursuits, such a priori knowledge is ignored or forgotten in the subconscious. In some cases, it is so forgotten that, upon being consciously introduced to it through various words and their socio-emotional baggage, it is not recognized as a priori. In such an event, it then easily is viewed as useless, trivial, or even arbitrary. This means that, for the most a priori kind of knowledge, it is difficult to prove to most people that is a priori, not because people lack it, but because it is only so subtly possessed in the first place in face of so many other, far more energy-intensive, and promising, concerns, such as: By what means may I fly at 40,000ft without either suffocating or freezing?
So, in regard to the most a priori knowledge, people can easily position themselves skeptically in regard to it: in face of empirical knowledge, and the ease of progress and reward in the relative mastery of the empirical realm, deduction can so easily be construed as to seem not to have any native (a priori) content, or, if it does, it is only trivial content. For example, if someone is determined that, in effect, deduction has no native content, then the only kind of reasoning by which someone else, of a deeply different metaphysics, can seem credible to such a person is if the reasoning which is offered is simply about everyday material things, those things' supposedly a priori ontological primacy, and the supposedly a priori epistemological primacy of the knowledge of those things. In short, a denial of truly a priori knowledge results in putting the epistemological cart in front of the horse, since the contents of the cart is so much more consumable and, thus, is so much more often 'in demand', than is the horse.