Differentiation--What are the maximum and minimum points

I understand how to work out the max. and min. points but I don't understand the meaning behind them, please explain

thank you

It's the highest or lowest point on the graph locally, or possibly globally.
Look at the graph of y = x^2
http://www.wolframalpha.com/input/?i=x%5…
As you can see, the minimum is at (0,0). It's the lowest point, as in it has the lowest y-value.

If it was y = -x^2
http://www.wolframalpha.com/input/?i=x%5…
Then we would now have a maximum at (0,0) which has the highest y-value.

A maximum point (or, more precisely a _local_ maximum) is where the graph of the line reaches a peak. A minimum point (or, more precisely a _local_ minimum) is where the graph of the line reaches a trough .... ie where the gradient changes.