Hitting the big red button stops the robot. A message will pop up on the screen. To get the robot moving again first rotate the red button in the direction of the arrows until it pops out. Then dismiss the popups and reinitialize the robot if necessary.
Singularities happen when the robot exceeds its safety limits while trying to move under hard constraints (e.g. "move in a straight line at a constant velocity").
Here is an example of a singularity in vertical motion:
And here is a singularity in the horizontal direction:
While driving the robot from the PolyScope UI on the pendant the most likely place to encounter this kind of problem is the "Move Tool" section under the "Move" tab:
Sometimes the robot hits things or itself or people (hopefully rarely). The UR5 is what is known as a collaborative robot - it is designed to be used in close proximity to people. This means that it incorporates safety features that minimize the risks associated with it unexpectedly hitting something. Every joint has force feedback and if at any moment the expected force exceeds safety margins the robot will come to an immediate stop. A popup will appear on the screen and it must be dismissed before the robot can continue operating.