What is meant by $d(xy)$? $dy/dx$ means derivative of $y$ with respect to $x$
But what is meant by $d(xy)$?
Where is the "with respect to term" here ? 
 A: $dx$ can be thought of as an infinitely small change in $x$, just as $\Delta x$ means a change in $x$ that is not infinitely small, and $dy$ would be the resulting infinitely small change in $y$.  Recall that
$$
\frac{dy}{dx} = \lim_{\Delta x\to0} \frac{\Delta y}{\Delta x}.
$$
Likewise $d(xy)$ would be an infinitely small change in the product $xy$. If $x$ and $y$ are both functions of $t$, then the product rule can be stated as
$$
\frac{d(xy)}{dt} = x\frac{dy}{dt} + y\frac{dx}{dt},
$$
and sometimes it is actually written as
$$
d(xy) = x\,dy + y\,dx.
$$
It can be rearranged into this:
$$
y\,dx = d(xy) - x\,dy
$$
and then both sides can be integrated:
$$
\int y\,dx = xy - \int x\,dy.
$$
In that form, it is called integration by parts.
A: It is the differential of $xy$ which is 
$$
d(xy)=x\,dy+y\,dx
$$
As for a somewhat intuitive interpretation, noticing that
$$
\Delta(xy)=(x+\Delta x)(y+\Delta y)-(xy)=x\Delta y+y\Delta x+\Delta x\Delta y\approx x\ \Delta y+y\ \Delta x.
$$
you can think of it as giving the increment in $xy$ in terms of the increments of $x$ and $y$, when those increments become infinitesimal (so that the second order terms become negligible). 
A: It is called the differential of $xy$, and denotes the linear approximation of the increment of $xy$ when $x$ and $y$ have small increments $h$ and $k$:
$$\Delta(xy)=(x+h)(y+k)-xy=\underbrace{xh+yk}_{\text{linear part in $h,k$}}+hk=\mathrm d\mkern1mu(xy)(h,k)+hk.$$
