A static character one which fails to change over the course of a text (play, short story, novella, or novel). This character remains the same, regardless of what happens around them (through their inability to change, denial of a problem, or ignorance).
One character that is static in William Shakespeare's play "Julius Caesar" is Caesar himself. This can be defined through his own dialogue found in Act 3, scene 1 when he exclaims,
But I am constant as the Northern Star,
Of whose true fixed and resting quality
There is no fellow in the firmament (lines 66-68).
Through Caesar's own admittance, Caesar is constant, refusing and unable to change. Like the North Star, whose position is a constant in the sky, Caesar admits that he is who he is and will not change. Other characters, based upon this, will always know what to expect from him (in the same way that they know where to look in order to find the North Star).
Also, any of the many characters who die in the play (without knowing much about them--flat characters) can be considered static. Not enough information is given about the character to define of the character changed prior to their death.