Said of a signal which, as it increases or decreases, the rate of change also increases or decreases. Such a signal, which viewed on a scope, looks like a curved shape. For instance, if the signal is exponentially increasing, it will start out as a fairly flat curve and then bend upward to an almost vertical shape. Also said of a device whose response to a linear signal is exponential. For example, some VCAs are switchable between linear and exponential response to control voltage; the latter is used mainly to provide snappier drum sound attacks.
Mathematically, not all curved signal or response shapes are, strictly speaking, exponential. However, synthesists tend to use the word "exponential" rather loosely to mean "not linear", whether the underlying mathematical function is actually exponential or not. Also, because of this loose usage, the word may not tell you exactly what the shape looks like. For instance, the signal may change slowly near zero and then "take off" when it moves away from zero, or it may do the opposite.