Plot twists take many forms: red herrings (purposefully misleading readers through story events), reveals, sudden, fortune-changing events, and many others. Here's a list of plot twists with examples from great movies. Just be aware, great plot twists are ALWAYS spoilers, so if you haven't seen the movie, skip the example (if you don't want spoilers).
Introduce Something New Probably the simplest way to twist the plot is to introduce a new character or situation. Something that shakes up the protagonist's world. It could be anything from a new kid on the street to a cancer diagnosis. Just remember that it's not a plot twist unless it's a game changer.
Example: Toy Story When Andy gets a Buzz Lightyear toy, life completely changes for Woody, pushing him out of his comfort zone and into situations he never thought he could handle.
Example: Monsters, Inc. When Sully finds Randall on the scare floor after hours Mike assumes Randall is cheating. Only later in the movie do Mike and Sully discover Randall's real plot, which is far more sinister than they ever suspected.
Example: Star Wars: The Empire Strikes Back Darth Vader reveals that he is Luke Skywalker's father. This really throws a spanner in Luke's goals to avenge his father's death and become a jedi because a) his father turns out to be the evil psycho he's been trying to kill, and b) Luke doesn't know whether or not he's inherited some of his dad's moral instability.
Example: Big Hero 6 Hiro loses his big brother Tadashi, someone he looks up to and depends on. With Tadashi gone, Hiro has to deal with his loss and learn to push through on his own.
Example #1: Up The story begins with a whole list of awful events for the protagonist, Mr. Frederickson. His wife dies, the big city has taken over his once-rural neighbourhood, he's going to lose his house, and he's being sent to an assisted living facility. Things are looking really bad when suddenly Mr. Frederickson releases thousands of helium balloons and floats his house away to South America, fulfilling his childhood dream of exploration.
Example #2: The Lego Movie Everything is awesome until Emmet finds the piece of resistance and thereafter gets arrested, becomes a fugitive, has to save the world, almost dies several times... you get the picture.