Princess is the feminine form of prince (from Latin princeps, meaning principal citizen). Most often, the term has been used for the consort of a prince or for the daughters of a king or sovereign prince.
For many centuries, the title "princess" was not regularly used for a monarch's daughter, who might simply be called "Lady" or a non-English equivalent. Old English had no female equivalent of "prince", "earl", or any royal or noble title aside from queen. Royal women, or women of the nobility, were simply addressed or referred to as "Lady" or "The Lady Firstname" for a particularly highborn woman.
As women slowly gained more autonomy and respect in European history, the title of princess became simply the female counterpart of prince; it does not necessarily imply being merely married to a prince, she may be the unmarried daughter of a monarch or other royal or noble of sufficient rank (king, queen, prince, grand duke, duke, or other). Further, a princess may be a hereditary sovereign, head of state in her own right.
The traditional meaning still applies in Europe in the sense that an untitled or at least non-royal woman who marries a prince will almost always become a princess with the possibility of eventually becoming Queen Consort, in nations where that title exists; but a man who marries a princess will almost never become a prince, unless his wife is expected to inherit the higher title of Queen regnant. If that is the case, then on her inheritance of that sovereign title, he will often become a prince (or remain one if he was already a prince).
In many of Europe's royal families, a king would grant his heirs actual or theoretical principalities to train them for future kingship or to give them social class. This practice has led over time to many people thinking that "prince" and "princess" are titles reserved for the immediate family of a king or queen. In fact, most princesses in history were not immediate members of a royal family but rather women who married into it.