A priori is Latin for what is before. In English, we use it to describe ideas, arguments, and assumptions that are based on conjecture, prejudice, or abstract reasoning rather than real-world experience. The opposite of a priori is a posteriori, which describes ideas that are based on experience.
A priori is a long-established loan phrase, so it’s usually not italicized. But it is italicized more often than other longstanding loanwords, probably because the a is easily mistaken for the English indefinite article.
Meanwhile, a one-word form, apriori, has gained some ground and now appears a significant fraction of the time (about once for every 20 instances of the two-word form in all 21st-century texts indexed in Google Books). It has given rise to several derivative words, including apriority (the quality of being derived in the mind instead of from experience) and apriorism (the use of a priori reasoning).
But when our correspondent proceeds to flying-machines, we have no longer the smallest taper-light of credible information and experience left, and must speak on a priori grounds. [Ralph Waldo Emerson, letter published in The Dial (1843)]
In fact the a priori reasoning is so entirely satisfactory to me that if the facts won’t fit in, why so much the worse for the facts is my feeling. [Erasmus Alvey Darwin in a letter to his brother, Charles Darwin (1859)]
All the knowledge that we can acquire a priori concerning existence seems to be hypothetical: it tells us that if one thing exists, another must exist, or, more generally, that if one proposition is true, another must be true. [Bertrand Russell, The Problems of Philosophy (1912)]
Typically, the person putting forth the rule assumes a priori that the rule is valid, and thereafter it seems that no amount of evidence or argument can change their mind. [Arrant Pedantry (2011)]
And can a socialist government ever adequately determine a priori the incentives a private company needs to justify the risk and effort of investing? [Financial Times (2012)]
They involve so much complexity and ambiguity that hypothesized solutions can rarely be “proven” right a priori. [Forbes (2012)]