If you mix water and water there should be no temperature change unless the two amounts of water are at different temperatures. If that is the case, the final temperature will be somewhere between the individual temperatures of the waters before they were mixed.
Acetone is the most common ingredient in nail polish remover and as you may know, it evaporates easily because the attractive forces between the individual acetone molecules are very low.
When you mix acetone with water, you are now mixing an organic molecule - acetone - with a polar molecule - water. There are also only weak intermolecular forces between the acetone and the water. Because of this, the acetone will evaporate from the water. When it evaporates it will take some heat energy from the water and the result is that the water will cool down. The same type of thing happens when you put alcohol on your skin. As the alcohol evaporates your skin cools down.