From the way your question is phrased, it seems as though you want to get a handle on particular distributions rather than the space of all distributions. In which case, the result cited by Debraj is probably the most comprehensive. Properly stated, the result is:

Theorem: If $T \in C^\infty(\mathbb{R},\mathbb{C})'$ (continuous dual) with $supp T \subseteq K$ ($K$ compact) then there are integers $n_1$, $n_2$, ..., $n_p$ and continuous functions $f_1$, $f_1$, ..., $f_p$ with supports in $K$, such that

$$ \sum_{j=1}^p f_j^{(n_j)} = T $$

The references for this are: Schwartz Theorie des distributions (1965) and Vo Khac Khoan Distributions, Analyse de Fourier. Operateurs aux derivess partielles (1972).

Then, of course, any arbitrary distribution can be written as the sum of distributions with compact support in a "nice" way.

From this point of view, the best examples are ones that are close enough to continuous functions that they are accessible (sorry, I know you're a category theorist but read that in British not categorish) but far enough away that you see some weird behaviour that you wouldn't expect if everything was a nice, continuous function. The examples mentioned in other answers are all good from this point of view: delta functions, derivatives of delta functions, $L^p$ functions, derivatives thereof. I'd add a few things like the Dirac comb, $\Delta_{a} = \sum_{n \in \mathbb{Z}} \delta_{n a}$ for $a \in \mathbb{R}$, $a \ne 0$, which has a particularly nice Fourier transform. You could integrate this to get an infinite staircase function (the floor function, that is). Indeed, any piecewise continuous function is actually a limit of a sequence of variations on the theme of Dirac's comb (i.e. where the tines can vary in length and separation) so Dirac's comb and its derivatives are the "only" distributions you need to know about.

But for me, this is the wrong way to think about distributions. If you want to understand distributions by looking at specific examples then you should really say that distributions are just smooth functions with compact support but in a slightly different topology. Once you've grokked the topology, then there's no reason not to simply think about really nice smooth functions. And if you haven't grokked the topology, then none of the "examples" is going to give you a good intuition as to how distributions behave. Indeed, I'd say that most of the examples are designed to make you think about the topology and to "shock" you into realising that the topology isn't what you naturally assume it should be when thinking about smooth functions.

I think of distributions simply as dual to smooth functions. The fact that we can think of functions as distributions is simply down to the fact that we have a pairing

$$ (f,g) \mapsto \int_{\mathbb{R}} f(t) g(t) d t $$

between many of the different function spaces that we can define. (Note the lack of conjugation.) This pairing defines a map from the one function space into the dual of the other and we can ask how much of the dual we can see in this way. That's essentially what the results about representing distributions try to answer. But this doesn't give much intuition as to what the dual space looks like as a whole because it tries to build it up piece by piece, each time saying "have we got it all yet"?

For example, many of the answers you got talk about differentiation of distributions. How do we know that we can differentiate these? In one answer, you got the formula $\partial \phi (f) = - \phi( \partial f)$. Where did that minus sign come from? After all, if I'm in tempered distributions then I can define the Fourier transform of a distribution and then the formula is $\mathcal{F}(\phi)(f) = \phi(\mathcal{F}(f))$. Why a minus sign on the one and not on the other? And I can multiply smooth functions, so why can't I multiply distributions? What's going on?

The truth is that by simply embedding functions into distributions you miss out on the whole duality story and the difference between defining a dual operator versus an extension operator.

But I've already written up this part on the n-lab so I'll simply refer you to there for the next chapter. Take a look over there. And while you're there, add your favourite of the above examples and correct the statement of the theorem.