Let's consider the graph of some function $y(x)$. I've just drawn some random curve:
We take the area under the curve as being made up of lots of little rectangles like the pink rectangle I've drawn. I've drawn the rectangle at the position $x$ on the $x$ axis so the height of the rectangle is $y(x)$. The width of this rectangle is $dx$, where by $dx$ we mean some tiny distance measured along the $x$ axis. We make $dx$ small enough that $y(x)$ can be taken as constant over the width of the rectangle. So the area of this rectangle is then just:
$$ dA = y(x)dx $$
Then the area under the whole curve is just the sum of the areas of all these tiny rectangles under the curve. Should you be interested in finding out more about this it is called Riemann integration.
Anyhow, so find out the physical significance of the area you just need to look at what $y(x)dx$ means. For example you ask about a graph where $y$ is the velocity and $x$ is the time. In that case the area of our little rectangle is:
$$ dA = v\,dt $$
And velocity $v$ times time $dt$ is just the distance moved at a velocity $v$ in a time $dt$. So the area is a distance moved. Add together all the little rectangles between two times $t_1$ and $t_2$ and we get the total distance moved between the times $t_1$ and $t_2$.
There are lots and lots of different graphs we could consider, but in all cases just think about the quantity $y\,dx$ and ask yourself what that quantity means. This will generally make it clear what the area under the graph represents.