I have come accross a lot of articles that talk about inverse problems. However, I dont really appreciate the uses due to my poor understanding of the notion.
From the mathematics point of view, when does a problem qualify to be called an inverse problem.
I will give an example from my field. Suppose you have a signal corrupted with additive noise: $$ y = x + w$$
You receive the noisy signal $y$. For to be any good use, you need to de-noise the received signal to get the approximation to the sent signal $x$: $$ \hat{x} = f(y) $$
by using some operation $f$.
This is essentially 'inverse problem' since you need to estimate the original signal from the corrupted one. Design of a good $f$ will reduce the risk or error in some sense, say mean square error:
$$ \mathbb{C} = \mathbb{E}|| \hat{x} - x||^2 $$
The one with lowest $\mathbb{C}$ wins the race. I hope you see how its different from the forward problem where you are given $x$ and you apply some operation to find a good $y$ in some sense.
To put it very intutively:
Forward: What will you do to get a certain type of observation?
Inverse: What happened at source which generated this type of an observation?
further explanation is on wiki as pointed out. here