Who introduced the notation $x^2$? In the book 'Problem Solving and Number Theory' I read 

The  law  of  quadratic  reciprocity  was  discovered  for  the ﬁrst
  time, in a complex form, by L. Euler who published it in his paper
  entitled “Novae demonstrationes circa divisores numerorum formae $xx + nyy$ .”

When and who introduced the notation $x^2$ ? What is the name for this notation? ( Not scientific, is it? )
 A: I don't know specifically who, but I recall that the notion was already invented during Euler's time.
It was just conventional to write $xx$ instead of $x^2$, i.e. one would write $x, xx, x^3, x^4, \ldots$.
This is probably similar to why we write $f',f'', f^{(3)}, f^{(4)}, \ldots$ for the notation of a derivative.
A: In their modern form, exponents were introduced by Descartes in the early $1630$s, at the same time as $x$.  There are numerous precursor forms of the exponent.
Although Descartes used the notation $x^n$ for $n \ge 3$, he ordinarily used $xx$ instead of our $x^2$. The notation of Descartes was fairly quickly widely adopted, with England as usual being more cautious.  The form $x^2$ was used by some people, the form $xx$ by others. Euler used both.  I believe he used $x^2$ far more often than $xx$. Maybe he thought $xx$ looked nice in a title.
A: According to this page the earliest known use of integers to represent repeated multiplication is by Nicole Oresme in the mid 1300s. However, he didn't use a raised integer notation. The rest of this answer is taken from that page.
Nicolas Chuquet used raised integers in 1484, though for him $12^3$ was a shorthand for $12x^3$.
In 1636 James Hume used roman numerals as exponents, e.g. for $12^3$ he would have written $12^\textrm{iii}$, but apart from that minor distinction he was essentially using modern notation.
Rene Descartes used raised arabic numericals as exponents in 1637, with the exception that he tended to write $xx$ rather than $x^2$, though he would still write $x^3$, $x^4$ etc. He wrote:

...$aa$ ou $a^2$ pour multiplier à par soiméme; et $a^3$ pour le multiplier encore une fois par $a$, et ainsi à l'infini.

which roughly translates as

...$aa$ or $a^2$ to multiply by itself, and $a^3$ to multiply again by $a$, and so ad infinitum.

