A logic gate accepts a number of binary inputs (usually two), and outputs another binary value. You can connect outputs to inputs to make new logic gates. A gate with which you can make any other gate is called a universal gate. One example is the nand gate which takes these two inputs, andand outputs One thing you can make with these components is an adder, which accepts two inputs and a carry bit (binary digit), and outputs a value and another carry bit, allowing you to chain them together. You can fidddle around with it here.
This device can be made out of nand gates. Here is a list.