A constant variable is a characteristic that in principle can vary, but that is kept constant for the sake of a particular experiment. Keeping some variables constant is crucial to scientific experiments, because it is impractical to conduct experiments at all possible values of all variables.
In a chemistry experiment involving chemical reactions of two substances, in principle you could conduct your experiment at any one of hundreds or even thousands of different temperatures. In principle, temperature is a variable. However, for the sake of a particular experiment, you may decide to hold the temperature constant, for example, at the boiling point of water. In such an experiment, temperature is thus a constant variable.
In a biology experiment involving the cultivation of beans, in principle you could conduct your experiment with any one of dozens or even hundreds of varieties of beans. In principle, variety of bean is a variable. However, for the sake of a particular experiment, you may decide to use only one specific variety of bean, such as the common green bean. In such an experiment, variety of bean is thus a constant variable.
In a psychology experiment involving performance at a motor task, in principle you could conduct your experiment using people from throughout the entire age range. In principle, age is a variable. However, for the sake of a particular experiment, you may decide to work with participants who are all the same age (within, say, a month or two), such as 3.5 years of age. In such an experiment, age is thus a constant variable.