You can think of a number as representing a quantity on a line.

    <--*---*---*---*---*---*---*-->      
      -3  -2  -1   0  +1  +2  +3

Numbers to the left of zero are negative, numbers to the right of zero are positive. They're different in the way a reflection is different; -2 is the reflection or opposite of 2.

Also, to work with numbers, you can think of positive ("+" or addition) as "move to the right" and negative (or subtraction) as "move to the left." And you always start at zero.

So if you have 1 + 1, you start at zero move right 1 and then move 1 more place to right on the number line (i.e., "you add 1") , and land at 2. So 1 + 1 = 2.

                   *--->--->      
    <--*---*---*---*---*---*---*-->      
      -3  -2  -1   0  +1  +2  +3

If you have 3 - 2, you start at zero, move 3 places to the right, then move 2 places to the left on the number line (i.e., "you subtract 2"), and you wind up at 1. So 3 - 2 = 1.

                       <-------*
                   *----------->    
    <--*---*---*---*---*---*---*-->      
      -3  -2  -1   0  +1  +2  +3

Now let's get tricky. If you have 3 + (-4), you start at zero and move 3 places to the right. The plus seems to indicate you're going to move to the right, but before you ever get a chance to move to the right you hit that pesky "-" sign, so you have to start moving to the left. You start at 3, and move 4 places to the left on the number line. You wind up at -1. So 3 + (-4) = 1.

               <---------------*
                   *----------->    
    <--*---*---*---*---*---*---*-->      
      -3  -2  -1   0  +1  +2  +3

Incidentally, that's the same as if you took the equation 3 - 4. Start at at zero, move 3 places to the right, then move 4 places to the left, and once again you're at -1.

It works on the other side of zero as well. Take -2 + 4. Start by moving two places to the left of zero. Then you hit the + 4, so you move four places to the right, winding up at 2. So -2 + 4 = 2.

           *--------------->
           <-------*
    <--*---*---*---*---*---*---*-->      
      -3  -2  -1   0  +1  +2  +3

Bam! Math!

P.S. You can also use the number line for multiplication and division, but that's another lesson.