An example of a concave shape is a crescent, such as a crescent-shaped roll. If you join the two ends of a crescent roll to form the shortest line, all points on the line are outside of the object, which is the roll. Concave shapes dent inward in at least two points, which when joined, form a line that falls partially or wholly outside the shape or object.
Simple examples of convex shapes include a circle and square. If you join two points in a circle or square to form a line, all points of the line remain in the object, which may be a ball or computer monitor. Convex shapes do not dent inward; however they may bulge outward, such as on an octagon or triangle.
In graphs or tables, a concave line describes the function of a variable. For example, when you place two points on a graph and connect them with a straight line, if no part of the line segment is above the graph, then the variable that the points represent is concave.
A convex line in a table or graph describes the function of a single variable. For example, when you place two points on a table and connect them with a straight line, if no part of that line segment is below the table then that variable is convex.