What Are Rests?
Items that appear in a measure that have no sound are called RESTS.

 
Rests are units of time in music where no sound is heard. You count them the same as notes.  For every note that exists, there is an equal unit of rest.  They take up the same amount of time in a measure as a note, but there is no sound.  You use them as you would in a measure, you just don’t play or sing. 
 
Here are the basic rests that you will often see:

whole rest (4 beats.)  It hangs from the 4th line, regardless of treble or bass clef, and looks like a "hole" in the line.

half rest (2 beats).  It sits on top of the 3rd line, regardless of treble or bass clef.

quarter rest (1 beat).  It splits the 3rd line, regardless of clef and looks like a couple of "z" 's on top of each other. 

eighth rest (1/2 beat).  It splits the 3rd line, regardless of clef, and looks like a little "7" or "y".

sixteenth rest (1/4 beat).  It splits the 3rd line, regardless of clef, and looks like 2 "7" 's or "y" 's on top of each other.


How do they work?
As said before, you use these in a measure just like you would notes.  Look at this example:

  The counts of this measure are 1-2-3-4.  Just like if there were a quarter note in place of that quarter rest.  The only difference in this case, is that on beat 3, you would not sing or play.

Here is one more example of rests, this time using eighth notes:

The counts of this measure would be 1&-2&-3&-4, just like if you had eighth notes where the eighth rests are.  The rests are on beats 2 and 3.  On the upbeats, or the “&”s of those notes, you would play or sing, but not on the downbeats.  You would rest there.
 
Now, just like notes can have dots, so can rests.  It is important to remember that the dot adds half of the original value to the rest, just like it does the note.  Dots will appear in the same place on rests as they do on notes, the right side.  Here is an example:

 Half rest = 2 beats of rest.   Dot = half of value, so 1 beat of rest.  2 beats + 1 beat = 3 beats of rest.

So the dotted half rest gets 3 beats of rest, just like the dotted half note gets 3 beats of sound.  See, not that hard is it?
 
Just like anything else, when dealing with rhythm, it is most important to make sure the measure is complete.  That means in 4/4, each measure has to have 4 beats.  Always count correctly and make sure your measures add up.