This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to understand it better, but I don't know of too many examples since I don't see too many people talk about them. What examples are common in your field, or what examples do you personally think are very revealing? Here's what I've got so far, starting with Terence Tao's example. Feel free to modify any of these examples if I'm not stating them correctly and to elaborate on them in answers if you want.
- $F_p[t]$ is a toy model for $\mathbb Z$.
- $F_p[[t]]$ is a toy model for $\mathbb Z_p$.
- Simplicial complexes are a toy model for topological spaces.
- $\mathbb Z/n\mathbb Z$ is a toy model for $\mathbb Z$ (for the purposes of additive number theory).
- The DFT is a toy model for the Fourier transform on the circle.
Which properties of the original objects carry over to your toy model, and which don't? As usual, stick to one example per post.