Lime water is the common name for saturated calcium hydroxide solution. The "lime" has nothing to do with the citrus fruit called "lime". Many calcium compounds are named "lime"; see also slaked lime and quick lime.
Etymology
Making lime water
Lime water can be made by adding calcium hydroxide to distilled water, The mixture needs to be shaken every few hours for a period of 24 hours to ensure that as much calcium hydroxide dissolves as possible. It is then left to settle and the solution is syphoned off the sediment.
Uses of Lime water
Lime water reacts with carbon dioxide to produce a precipitate of calcium carbonate. It is used as a test for carbon dioxide.
When the old vaudeville style medicine shows promoting various patent medicines were in full swing, lime water was often used as a part of the act. The salesperson would have an audience member blow through a straw into a glass of lime water. Since his exhalations contained carbon dioxide, the water would cloud; the huckster then announced that this reaction proved that the audience member suffered from some ailment.
The huckster had a patent medicine bottle filled with vinegar or some similar acid. He then would pour some of the acid into the glass of cloudy limewater. The acid reacted with the calcium carbonate, and the water would instantly clear. This demonstrated the potent effect of the nostrum he was selling to eliminate the "disease" demonstrated by the reaction.