It is primarily a mathematical term. The prefix "a-" means without or not. The root word means to meet. So fundamentally, it means not to meet.
Consider a curve such as y = 1 / x
As x increases, y decreases. The positive x-axis is an asymptote of the curve. As you continually increase x towards infinity, the value of y gets closer and closer to zero, and hence the curve gets closer and closer to the x-axis but never reaches it. We say the curve is asymptotic. Other regions of this curve are asymptotic as well, but no need to go into further detail.