Graphs are useful because many people prefer looking at pictures to looking at equations. They're not necessarily better, though. While a picture may be worth a thousand words, it's really only worth one equation.
When giving a presentation, a graph makes a very nice visual. Different sorts of graphs show up in real life, from president approval rating surveys, to financial market summaries, so it's good to be comfortable reading them. Slip into your sock monkey slippers, stoke the fire and settle in for an evening of satisfying graph-reading.
Graphs let us see patterns in numbers that we might not be able to see merely by looking at a list. Lists are nice some of the time, but are we running out to the store for a gallon of milk right now? We didn't think so.
We can look at the relation generated by the equation x = y, or we can look at a graph of a straight line. Lots of things in real life, such as population sizes or the height of something thrown into the air, can be measured to generate graphs with familiar shapes, including straight lines, quadratic functions, and exponential functions. Remember last week when you never would've counted quadratic functions or exponential functions among a list of "familiar shapes"? Move over, triangles and squares. There are some new shapes in town.