In high school, you probably learned that equations, such as y = 2x + 1, can be represented by a graph. This is done by plotting the points (x, y) which satisfy the equation. For example, the point (1, 3) is on the graph of the equation y = 2x + 1 since (3) = 2(1) + 1. This is illustrated below.

Here are some graphs that are usually introduced in a course in college algebra or analytic geometry.

Here are some other graphs.

Notice, the last two graphs have sharp corners, whereas the first four graphs are all smooth.  A singularity is a point where a graph has a sharp corner.  Looking at the last two graphs we see that the first graph intersects itself at the point (0, 0).  The second graph comes to a sharp point at (0, 0).  In both situations there is a sharp corner at the point (0, 0), so we say that both of these graphs have a singularity at (0, 0).

You can also have singularities along entire lines instead of just at single points.  For this we need to look at graphs in three dimensions.