Truth tables are a way of writing down all the possible combinations of statements and saying whether the whole combined statement is true or false.  Here they are for the five connectors:

The OR statement in this case says something like, “Well, if statement A is true OR statement B is true, or they both are, then the overall proposition is true.” So the only time the whole statement is false is when both A and B are false.

The AND statement is more fussy – both A and B have to be true before the whole combined statement is true.  All other cases end up with it being false.

The first two rows are the only ‘legitimate’ ones for the IMPLIES connector.  If A is true and the thing it implies is also true, then the whole thing is true.  If A is true but the thing that it implies doesn’t actually happen (i.e. isn’t true), then the whole statement’s false.

The shaded rows are for when the first part of the statement is false – in this case we can’t really say anything about the overall validity of the statement, so we give it the benefit of the doubt and say it’s true.

Equivalence is basically looking for whether the trueness or falseness of the two statements matches.  If they match, then the whole statement’s true.  If they don’t match, then the whole statement’s false.

NOT’s pretty simple. It reverses the statement’s true or false value.