It's actually much simpler to understand the meaning of eccentricity with use of formulae, defining corresponding conic section. Say, in polar coordinates, r=l /(1 - e cos θ), e being the eccentricity. From this it's apparent how the distance from the focus to the point on the curve changes depending on polar angle θ -- this distance does not change at all for a circle, changes between certain lower and upper bound for an ellipse (and the more e differs from zero, the more "eccentric" it gets, i.e. the more the difference between r(θ = 0) and r(θ = π) ), r becomes infinite for θ = 0 for a parabola (with e=1), and finally r becomes infinite for θ = ±θ0 for a certain non-zero θ0 for a hyperbola (with e>1), the larger the e the farther θ0 from 0.

Without formulae -- it becomes more of a speculation, as eccentricity in the meaning as in your question is only defined for conic sections, how quantitatively it affects the shape can only be understood when one equation or another for the curve in question is used. As for a speculation (it can be obtained from the consideration in the paragraph above) -- circle is "centric", its eccentricity is zero, as all its points are at the same distance from its center (coinciding with both foci). For an ellipse, a squeezed circle, foci no longer coincide with its center, the more the e, the farther apart a focus from the center gets. Namely, e = c/a, where c = linear eccentricity = the distance between the ellipse center and one of its foci, and a = its big semi-axis = the largest distance between the center and a point on the ellipse (such largest distance achieved at θ = 0 or π). The more the e, the farther the foci from the center, the more "eccentric" the ellipse is, the more it differs from a circle (gets more squezed along the short axis). Parabola is a limiting case of an ellipse when e becomes 1, the "center" of the curve goes to infinity, making effectively c = a (both of these infinite, while "a-c" = r(θ = π) staying finite). b/a goes to zero, b being the small semiaxis of the ellipse (whose limiting transition to a parabola is considered). Hyperbola is obtained when b is considered imaginary, with certain geometric intuition. Or in projective geometry -- as a further evolution from parabola, the curve center going past infinity back to a proper plane point from the other side (relative to where it was in the starting ellipse of this transformation).