According to Nocedal & Wright's Book Numerical Optimization (2006), the Wolfe's conditions for an inexact line search are, for a descent direction $p$,
Sufficient Decrease: $f(x+\alpha p)\le f(x)+c_1\alpha_k\nabla f(x)^T p$
Curvature Condition: $\nabla f(x+\alpha p)^Tp\ge c_2 \nabla f(x)^T p$
I can see how the sufficient decrease condition states that the function value at the new point $x+\alpha p$ must be under the tangent at $x$. But I'm not sure what the curvature condition is telling me geometrically. Also, why must the relation $c_1<c_2$ be imposed? What does this accomplish, geometrically?