There are several different ways to compute this sum. We will present two approaches here.
$\underline{Use\ basic\ formula}$
If the number of numbers in each term is limited (i.e. in this case, it is $2$), it is possible to convert such a sequence to the sum of power sequence. Because $$n(n+1) = n^2 + n$$ we can rewrite the original sequence as
$$ \begin{align*} & 1\times 2 + 2\times 3 + \cdots + 19\times 20 \\ = & (1^2 + 1) + (2^2 + 2) + \cdots (19^2 + 19) \\ = & (1^2 + 2^2 + \cdots + 19^2) + (1 + 2 + \cdots + 19)\\ = & \frac{19\times (19+ 1)\times(2\times 19+ 1)}{6} + \frac{19\times (19+1)}{2}\\ = & \boxed{2660} \end{align*} $$
$\underline{Use\ Combinatorial\ Identity}$
Alternatively, such expression can also be computed using the Hockey Stick Formula as show below:
$$\begin{align*} & 1\times 2 + 2\times 3 + \cdots 19\times 20 \\ = & (1\times 2)\Big(\frac{1\times 2}{1\times 2} + \frac{2\times 3}{1\times 2} + \cdots + \frac{19\times 20}{1\times 2}\Big) \\ = & 2\times\Big(C_2^2 + C_3^2 + \cdots + C_{20}^2\Big) \\ = & 2 \times C_{21}^3 \\ = & \boxed{2660} \end{align*}$$
The simplification of the combinatorial series relies on the Hockey Sticker Identity (See # 4139).