An asymptote is a line that a curve approaches but never touches. Linear graphs, such as y = mx + b, do not have asymptotes because they extend infinitely without restriction on the x or y-axis values. However, for functions like y = 1/x, there are two types of asymptotes: vertical (y-axis) since x cannot be zero and horizontal (x-axis) since the fraction's numerator prevents it from equaling zero. These concepts become increasingly important in advanced mathematics.