The short answer is not very much.  A part per trillion, as it applies to drinking water constituents, is a measure of concentration - how much contaminant there is in a given volume of water.  One ppt means that there is one gallon of contaminant per trillion gallons of water or more specifically 8.34 pounds (weight of 1 gallon of water) of contaminant per trillion gallons of water.  It is an exceptionally small amount and is probably easier to visualize by the following examples.

1. The average distance between the earth and the moon is approximately 240,000 miles or a bit more then 15.2 billion inches.  One trillionth of this distance is 15 thousands of an inch.  This is about the diameter of a human hair.
2. Presume the average lifespan was 75 years or about 98.7 million seconds.  One trillionth of this amount of time is approximately 2.4 milliseconds or thousandths of a second.  The average eye blink takes 100 to 400 milliseconds.
3. The surface of the earth, including land and water, is about 197 million square miles.  One trillionth of this is about 5,500 square feet which is roughly equivalent to the footprint of the Town Hall, prior to renovation.