In an interval in which a graph is concave up, the slope of the tangent lines to the graph are increasing (i.e. f''(x)>0). And in an interval in which a graph is concave down, the slope of the tangents lines are decreasing.
As posted the second derivative itself can tell you if the function is concave up or down, but the second derivative test can tell you f(c) is a local maximum or minimum. As stated if f is concave up (f''(x)>0) near c, then f lies above its horizontal tangent at c. Therefore, f(c) would be a local minimum, and similarly if f was concave down near c.