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:

    #
    ###
    1##

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.

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I was looking at the second part of the question first, interesting higher-dimensional examples.  Here are some non-isomorphic local rings $R$ (not necessarily algebras) that have length 2 or 3 and such that $R/m = \mathbb{Z}/p$:

Length 2:

 1. $(\mathbb{Z}/p)[x]/(x^2)$
 2. $\mathbb{Z}/p^2$

Length 3:

 1. $(\mathbb{Z}/p)[x]/(x^3)$
 2. $(\mathbb{Z}/p)[x,y]/(x^2,xy,y^2)$
 3. $(\mathbb{Z}/p^2)[x]/(px,x^2)$
 4. <s>&nbsp;$(\mathbb{Z}/p^2)[\sqrt{p}]$&nbsp;</s> $\mathbb{Z}[\sqrt{p}]/p^{3/2} = \mathbb{Z}[x]/(x^2-\lambda p,x^3)$
 5. $\mathbb{Z}[x]/(x^2-\lambda p,x^3)$ where $\lambda \in \mathbb{Z}/p$ is not a square. (Noted by Jonathan Wise.)
 6. $\mathbb{Z}[x]/(x^2+2x,4)$ (Similar idea to previous, in characteristic 2.) 
 7. $\mathbb{Z}/p^3$

(**Edit:** My notation for #4 was not strictly correct.)

<s>I *think*, although I can't really speak with authority, that these are all of them.</s>  I thought that I knew all of these rings, but that was naive.   One point is that among algebras over $\mathbb{Z}/p$, the length is too small to see anything non-toric.  But you can also have local rings that look like these toric local algebras (which I listed first), but have carries.  The most creative one is the fourth one of length 3, namely $(\mathbb{Z}/p^2)[\sqrt{p}]$.  You can express an element of this ring as three digits in base $p$, say $d_2d_1d_0$.  Then addition carries from $d_0$ to $d_2$.

I would also guess that all of these generalize to $\mathbb{F}_q$, using [the Witt vector][1] construction in the cases with carries.  And maybe it is again all of them.

  [1]: http://en.wikipedia.org/wiki/Witt_vector