What is a delta potential?

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.