The definition of nilpotent Lie algebra $\mathfrak{g}$ says that its lower central series terminate at the zero subalgebra.
The lower central series is the $\mathfrak{g}$, then scalar product of $\mathfrak{g}$ and $\mathfrak{g}$, then $\mathfrak{g}$ product with the former etc. etc. (The definition can be found on Wikipedia.)
If some member of this series is zero, then the Lie algebra is nilpotent.
My question is, what is the interpretation of the lower central series terminating to zero?
I only know nilpotence of matrices, where the matrix on the power of some positive integer is zero. How to connect the various definitions of nilpotence - for groups, matrices, algebras etc.? Why do we need this definition and what does it saying in layman terms about the given structure?