A connection point is any part of a LEGO element that allows for a secure connection to another. Stud and Tube is the basic type of connection and was part of the patent for the bricks still used today.
The number of connection points has grown over the years and encompasses both approved and illegal connections.
Clutch is a term used to describe how strong a connection is, ie. the force required to separate two elements. Sometimes described as clutch power (not to be confused with the character in The Adventures of Clutch Powers) the strength of the connection can depend on several factors including:
- the type of material used (ie. ABS or cellulose acetate)
- Ultra violet (sun) damage or age 
- part design (ie. clip vs brick)