The terms "paradox" and "oxymoron" both refer to literary terms involving contradictions that reveal a deeper truth. The Merriam-Webster dictionary defines "oxymoron" as "a combination of contradictory or incongruous words," whereas the dictionary defines "paradox" as "a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true." A person, item or situation can also be described as a "paradox" if he or it exhibits this combination of truth and contradiction.
The difference between a paradox and an oxymoron lies in the length of the phrase or term being described. Oxymorons are short, two-word phrases, presented without further explanation. A paradox can be made up of a longer phrase, a paragraph, or even a full story. Basically, "paradox" refers to the larger idea behind a contradiction that illuminates a truth, whereas an oxymoron is the term for a simple two-word phrase that illustrates the same idea.
Because of the strict rules that govern oxymorons, they are simple to illustrate. One common example of an oxymoron is the phrase "jumbo shrimp." The word "jumbo" means large and the word "shrimp" translates to small, yet the phrase makes sense. Another clear example is "open secret." Technically a secret cannot be well-known, but when those two words are used together they make sense of a concept that is seemingly contradictory.
Since paradoxes are concepts that can't always be summed up in a neat phrase, examples are less obvious. One clear example is George Bernard Shaw's quote "What a pity that youth must be wasted on the young." On first glace, the statement is contradictory, but on further examination it reveals the truth that young people often do not appreciate the value of their fleeting youth. Another common example is Shakespeare's line in Hamlet: "I must be cruel only to be kind." This illustrates a contradictory concept that speaks to the value of doing what is best for someone, even if it initially seems harsh.