What is the definition of an extremum of a function? In calculus extremum is a common topic, but I don't understand what it is. What is the use of this extremum, especially in practical life?    

In analysis of differentiable real-valued functions of a single argument, an extremum is usually defined to be a place where the derivative is zero.
$$ x_0 \text{ is an extremum of } f(x) \text{ iff } f'(x_0) = 0 $$
To identify an extremum as a local maximum, minimum or an inflection point, you look at the second derivative:
$$ x_0 \text{ is a local maximum of } f(x) \text{ iff } f'(x_0)=0\text{ and } f''(x_0)<0  $$
$$ x_0 \text{ is a local minimum of } f(x) \text{ iff } f'(x_0)=0\text{ and } f''(x_0)>0  $$
$$ x_0 \text{ is an inflection point of } f(x) \text{ iff } f'(x_0)= f''(x_0) = 0  $$
