Turn an LED off and on with the same push button
Assuming you don't want to do this with a simple microcontroller, I'd recommend you an easy logic gate circuit as from here:



Imagine Vout is low for a start. This will make the first gate (G1) output high. That makes the second gate (G2) output low, so this situation is stable. 

Now, someone presses the button. That connects the high output of G1 to its input, so that the output becomes low. The low on the inputs of G2 makes the output of G2 high. We're in a stable situation again.

When someone presses the button again, the same happens, but in the opposite way: the low output of G1 is connected to its input, making the output high. That is connected to the input of G2, making the output of G2 low. We're back to the first situation.