Basic Gates

The core of breadboarding, and circuit building is based off of AOI logic.  AOI stands for AND, OR, INVERTER.  These are the most basic gates to use.  These gates all have different truth tables and outputs depending on their input.
The AND gate outputs a 1 only when all of the inputs are a 1, and outputs a 0 when only one or none of the outputs are a 1. When inputs are put through an AND gate, the result is written by multiplying the inputs together.  The truth table is as follows:
X   Y    F
0    0    0
0    1    0
1    0    0
1    1    1



The OR gate outputs a one when any of the outputs are a 1, or when all of them are, and outputs a 0 when all of the outputs are a 0. The result of inputs put through an OR gate are written with an addition sign written between the inputs.  The truth table is as follows:
X   Y    F
0    0    0
0    1    1
1    0    1
1    1    1

The NOT gate, also known as the INVERTOR gate, outputs a 1 when the input is 0, and outputs a 0 when the input is 1. This gate is particularly special because it is the only basic gate that only has one input.  (AND and OR gates can have two or more, unless one was to tie the inputs together) The truth table is as follows:

X    F

0    1

1    0

Click Here for more information on the gates and a few more complex ones.