What is a delta potential? I know how a delta potential is described mathematically but how can a delta potential be a 'well'? Does it have particles outside the 'well' and 'bind' it or does it somehow have particle inside it? 
Also, if we consider three dimensions then won't the potential be present at only one point in space meaning there being no particles inside it?
For example, suppose a situation where at origin we have a delta potential. Now, if a particle is at say the right side of the origin and moving towards it then how will it interact with the potential? Is this interpretation of the potential even correct?
I am sorry if I am unclear but I really don't understand the concept of delta potentials.
 A: I think that a helpful way to think of delta potential (and maybe on the delta function in general) is through a limit process: we start with a finite square well of width $a$ and depth $U=\lambda/a$, and ask ourselves "what happens when we take $a\to 0^+$?" This can happen when we are interested, for example, in scales that are much larger than the width of the well, so we want to somehow make an approximation to zeroth order in $a$, but still keep the effects of the well. A nice thing here is that there are many different limit processes that can lead all to the same result, which is a very general expression of the potential as $\lambda\delta(x)$.
Note, that even though the width of the delta function is zero, it still has effect as it has non-zero measure $\int\! dx \delta(x) = 1$, which is quite obvious from the limit process that we introduced. Because we make sure to make it deep, we still keep its effects on whatever comes near it.
A particle can be "trapped" in the sense that any finite potential well can trap a particle - it has a probability to be found outside the well, as its the wave function decay exponentially outside the well for $E<0$. Now the particle has higher probability to be found near $x=0$ than far from it, in contrast to a free particle which spreads throughout the entire volume.
A: Usually $\delta$-potentials are a mathematical model of interactions with very short range.
The advantage of such models is that on one hand they are "exactly solvable", e.g. the spectrum and eigenvectors are explicitly known, on the other hand many interesting physical features are retained despite the simplification involved in approximating short-range with zero-range.
For example, ionization processes can be suitably well described, at least to some extent, by time-dependent point ($\delta$) interactions.
