The water drops on the outside of the cup come from water that's already in the air. Water can be a solid (ice) liquid, or gas. As you increase the temperature, ice melts into water. Then as you increase the temperature more water evaporates into steam. Likewise, if you cool steam, you get water, and if you cool water enough you get ice. When we think of water as a gas we normally think of steam, which is very hot, but it can also be a gas a normal temperatures. There's always some amount of water as a gas just hanging out in the air, along with nitrogen and oxygen. (When people talk about humidity, they're talking about the amount of water vapor in the air.) So if you put a cold cup down on a table, the water vapor in the air that comes in contact with the cup gets cold and will turn into water. This process is called condensation.

The drops of water on the outside of a cold cup come from the air.Believe it or not, there is a significant amount of water vapor in the air that we breathe. This is why we have humidity. A cold cup can take thermal energy out of the water vapor in the air next to the cup, causing the water vapor to condense into liquid water. The same kind of phenomena occurs when you take ice cream out of the freezer - you see frost on the outside of the container. In this case the cup (or ice cream container) is below the freezing point of water. Therefore, the water vapor from the air will freeze when it comes in contact with the container, forming a layer of frost.

The water comes from the surrounding air. You see the amount of water vapor that can dissolve in air is a function of temperature. Now the amount of water that can be dissolved in air is smaller when the temperature drops, so if you have a cup filled with something cold, the outside of the cup is at a low temperature; this chills the air that is adjacent to the cup and since the chilled air cannot keep the water dissolved in the air, the water condenses.

As you know, water can behave differently - be in the gas, liquid or ice form. Water "decides" in which stage to be mostly because of temperature. So if the water is in gas form in the air suddenly gets somewhere where its too cold to be a gas, it turns in to water. If in that moment there is something hard around, it will condensate on it, if nothings, we will get a "raindrop" that just falls down. When you have a cold enough glass in the warmer air, the water from the air around turns into drop and condensates on the glass if it gets close enough to the cold glass. The air around contains up to 30 grams of water (in vapor) per cubic meter of air. More than enough to make your cup wet. Another question is if your cup has a drink in, how much of the water that condensates on the cup and was in the air before came to the air from the drink?

Air has water vapor in it, and the amount of water vapor that is stable in the air depends on temperature. Cool it off to where there is too much water vapor, and the vapor will condense out to form droplets, either on a cold surface, or just in the air (that's what clouds are). Heat the air or the water to the point where any amount of vapor in the air is stable, and the water will boil.

The drops come from the surrounding atmosphere. Because colder air causes condensation at a higher rate that warm air, water vapor contained in the surrounding warm air condenses out when it comes in contact with a colder cup.