A shape is concave if a line can be drawn between any two points in the shape that is not completely contained within the shape. ^{[1]}

Figure 1 is an example of a concave shape. Since it is possible to pick two points within the shape and draw a line between them that leaves the shape, the shape is convex.  

Figure 2 is an example of a convex shape. Note that the entire line is contained within the disc. 
The earth is often referred to as a sphere. However, it is not a perfect sphere. If you draw a line between the tops of two mountains, is the line completely contained within the earth? Is the earth concave or convex?
The line is not completely contained within the earth, so the earth is concave.
A bear wanted a nice cozy cave in which to spend the winter. She found a rock sticking out of the ground, but it vexed her that the rock was convex. She continued to look. Then she found a cave. It was concave, so she crawled in and went to sleep for the winter.

A curve is concave up over an interval if it opens upwards over that interval. A line segment drawn from any point to any other point on the interval a curve is concave up will be entirely above the line. A curve is concave down over an interval if it opens downwards. A line segment drawn from any point to any other point on the interval that is concave down will be entirely below the line. The point at which a curve changes from concave up to concave down is called the inflection point. 
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