Extremum is called maximum or minimum point of the function. Look at the picture of some function:
From the plot, one can conclude that the points (x1, y1), (x3, y3) are maxima of the function. The points (x2, y2), (x4, y4) are minima of the function. Both, these points are called extrema of the function.
The tangent of the function at extrema points is parallel to the abscissa axis (geometric sense). Hence, the derivative of the function equals to zero at extrema points (requirement of the extrema). In addition, derivative may not exist in extrema points.
Sometimes, we need to find minimal (maximal) value of the function at some interval [a, b]. In this case, one need to find all the extrema points which belong to this intervals and also check the values of the function at the borders of the interval.