Signal and system convolution
This is known as deconvolution. A typical approach would be to apply a Fourier transform to both sides
$$
y = x \circledast h \overset{\mathscr{F}}{\Leftrightarrow} Y=XH
$$
and divide both sides by the known $X$ to obtain $H=Y/X$. Then apply an inverse Fourier transform to both sides, yielding $h$. 

The linked wikipedia article goes into more detail on variations of this approach.