Wikipedia says Toy model (physics):
"In physics, a toy model is a simplified set of objects and equations relating them so that they can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory.
In "toy" mathematical models, this is usually done by reducing the number of dimensions or reducing the number of fields/variables or restricting them to a particular symmetric form.
In "toy" physical descriptions, an everyday example of an analogous mechanism is often used to illustrate an effect in order to make the phenomenon easier to visualize."
I'm not used to physics terminology: for example, if I remember good, a Physical Model (like the Standard Model) is not exatly the same as a Mathematical Model (from model theory)... and this is enough to make me suspicious about the meaning of the term "Toy Model" when used inside a mathematical framework.
$\mathcal Q 1$ The wiki page seems to talk about Toy model in Physics... so what is the exact definition of a Mathematical Toy Model?
and.. since the best way to understand are the examples
$\mathcal Q 2$ I'm looking for a big list of Toy Models, why they are introduced and how they are used.
About the closure votes:
unclear what you're asking : The question is really clear. Definition and list of examples are asked.
Too broad: "There are either too many possible answers" this is obvious since the Tag "Big List" is used (and is used with "soft question" tag)...", or good answers would be too long for this format." obviously wrong, as Mariano Suárez-Alvarez's comment shows.