When you add, you join two or more things together. If you add on to your summer cottage, you build an extra room that's connected to your existing house.
If your boss adds ten dollars to your paycheck, she gives you the money in addition to, or on top of, your salary. And when you add a column of numbers, you combine each one in the list until you get a final answer. You might verbally add something by making one last statement: "I'd like to add that I can't wait to visit again!" The Latin root is addere, "add to, join, or attach."