Rays in geometry are parts of a line that have a distinct endpoint on one side, but go to infinity on the other side. Therefore, opposite rays start from the same spot and continue infinitely to opposite directions (for example: up and down, left and right. All points on either of the two opposite rays are "collinear" (lying on the same straight line), even though the two rays don't constitute a single straight line.
You can use opposite rays to make an axis spanning on both sides of the number zero. For example, on a Cartesian coordinate plane, the x-axis (horizontal line) consists of the rays starting from zero and pointing to the right (positive side) and the ray starting again from zero and pointing to the left (negative side). As such, opposite rays are useful when you want to depict negative and positive values of the same measurement unit (temperature in Fahrenheit degrees for instance).
An angle consists of two rays that begin from -- or meet at -- a certain point and form a "vertex." Since opposite rays have a common point, they too form a vertex: the so-called "straight angle" (180 degrees angles). Opposite rays cannot form any other type of angle (acute, obtuse, right, reflex) apart from a straight one.
To avoid confusing two opposite rays with a single straight line, form a small ball with your pencil between them. Then draw the point of an arrow on both lines to depict the direction at which they are looking. As long as the lines end with an arrow and not another ball, it is implied they go on infinitely. An easy way to describe opposite rays to children is by using a clock pointing at 6:30 or 9:15, or a compass with different colors for the sides of its magnet.