Without knowing the exact detailed of your child's science project, I'll use an example to explain these variables. For instance, your child's question might be, "How does the amount of time you soak a piece of potato in grape juice change its color?"
Your child's guess or hypothesis might be that the potato will turn a darker color the longer it soaks in grape juice.
To answer this question and test the hypothesis, you would create an experiment using 3 or 4 cups of grape juice with slices of potato in them.
The manipulated variable is also called the independent variable. This is what you change. For this experiment, time is the manipulated variable. You would keep each slice of potato in its cup of juice for increasing amounts of time. For instance, cup 1 - 1 hour, cup 2 - 3 hours, cup 3 - 12 hours, cup 4 - 24 hours.
The constant is also called the controlled variable or simply the control. This is what does not change in the experiment. For instance, you would want to use the same amount of juice in each cup. You would want each cup to be the same size. And you would want the potato slices to be about the same diameter and thickness and come from the same potato. The constants are important, because it is very hard to test your hypothesis if you change more than one thing... Was the potato slice in cup 1 a lighter color because it soaked in the juice less time, or because it was much bigger than the other slices of potato?
The responding variable is also called the dependent variable. In this experiment, the color of the potato slice at the end of its soaking time is the responding variable.
For a simple experiment, such as this, it is best to have one manipulated variable, one responding variable, and everything else contant. Then it is easy to test the hypothesis and reach a conclusion - e.g., the hypothesis is proven true: the longer a potato slice is soaked in juice, the darker it gets. Or it gets darker up to 12 hours but no darker after that. Or it has to soak at least 3 hours to get darker.
A really good detailed explanation of variables can be found on Science Buddies, along with lots of free science project ideas with step-by-step instructions, AND help on their forum: