The delta well is the limit of the finite potential well, and results from decreasing the width of the well (obviously, sorry) but increasing the potential (previously constant in the finite well), as a consequence of that limiting process.
Consider a potential V (x) = -$\delta(x/\alpha) $
Although the spike is not physically feasible, (there are no infinities in nature) it's handy as a guide to some physical situations:(it might even represent the electron!, but perhaps not).
To be more realistic, it can represent cases where a particle is allowed to move in two regions of space, separated by a (very thin) barrier.
Since -$\delta(x/\alpha) $ is infinite at 0, irrespective of the value of $\alpha$, $\alpha$ may be considered to give an indication of the strength of the potential in a way set out below.
In fact, $\alpha$ does appear in reflection and transmission coefficients, as well as energy level estimates.
The delta-function well has just one bound state, and the energy inherent in that state is dependent on $\alpha$, the greater the value assigned to $\alpha$ is, the more negative energy results. In other words $\alpha$ acts as an indicator of the depth of a stationary state particle.
Obviously the amplitude of the location of the particle reaches its peak at 0, and then, depending on the strength of the delta function, falls away exponentially on both sides.