There are a bunch of different notions of length/depth in ring theory: Projective length, Artinian length, local depth, etc. If we take length to mean Artinian length, then Charles is right: The Artinian length of a finite-dimensional commutative algebra is just its dimension. Every such algebra is a direct sum of local ones, and you can chip away at each local summand of the ring from the bottom end, one dimension at a time.
The local algebras that have a description that looks as nice as $\mathbb{F}[x]/(x^n)$ are the toric ones. These local algebras are $\mathbb{F}[\vec{x}]$ divided by an ideal generated by monomials and a basis of monomials. You can make a diagram of the exponents of the monomials that aren't killed by the ideal. If the ring has $n$ generators, then the diagram is a stable stack of blocks in the $n$-dimensional orthant. For example, the algebra $\mathbb{F}[x,y]/(x^3,x^2y,y^3)$ has a basis of seven monomials: 1, $x$, $x^2$, $y$, $xy$, $y^2$, $xy^2$. The diagram of these monomials looks like this:
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I have put a 1 at the corner in the diagram corresponding to the monomial 1.
What is easy to forget is that all finite-dimensional local algebras with one generator in $m/m^2$ are of this form, but with more variables these are just special examples.