Stances are poses that some characters can take.
For the most part, stances are special modes that are only available to some characters (with a few exceptions, like the Stand system in the JoJo games that gives it almost all of the characters).
When in a different stance, you will be able to perform some different moves; likewise, some other moves will be unusable unless you switch to the proper stance to perform them.
Many stances leave you unable to block, or at a disadvantage somehow.
Some stances undo themselves after a certain amount of time, and some are permanent (i.e., you can’t go back, or not easily at any rate).
There are many cases where using stances at the right times open up an avenue to powerful combos.
Super special moves and MAX modes and such that require the expense of the power meter, as is common in 2D games, temporarily allowing you heightened abilities are not generally considered stances.
Simply having stances makes a character a bit complicated, so it can be a pretty good sign that a character is aimed at advanced players.