The eccentricity of a hyperbola is equal to:

A circle has an eccentricity of zero, so the eccentricity shows you how "un-circular" the curve is. Bigger eccentricities are less curved.

Different values of eccentricity make different curves:

At eccentricity = 0 we get a circle

for 0 < eccentricity < 1 we get an ellipse

for eccentricity = 1 we get a parabola

for eccentricity > 1 we get a hyperbola

for infinite eccentricity, we get a line