If you are asking about radio telephony then,
b. A pilot who encounters a Distress condition should declare an
emergency by beginning the initial communication with the word
“Mayday,” preferably repeated three times. For an Urgency condition,
the word “Pan-Pan” should be used in the same manner.
- There is Pan Pan which is covered here, although that is typically used for an "urgent" situation.
- Primarily there is Mayday for emergencies.
- There is simply "I have an emergency...." (or if it’s potentially less urgent “a problem” r.e. Apollo 13) however this is not the preferred method.
Those are the three big ones and will get the attention of a controller as well as alerting other pilots on the frequency that there is a problem in the area.
As for the other part of your question, the light signal chart you linked is for tower to aircraft communications when radio is not available. These signals are only used when the aircraft is in close (visual) proximity to the tower.