A not-$\omega$-saturated model. I'm new to $\omega$-saturated model and the likewise and although I'm aware of examples of $\omega$-saturated models $(\mathbb{Q},<)$, I can not really imagine a not-$\omega$-saturated model and how to prove this absence of the property.

The standard example is probably the natural numbers $(\mathbb{N}; +,\times,<,0,1)$ (or a variant thereof). The set of formulas $$\\{x>1, x>1+1, x>1+1+1, ...\\}$$ (which uses no parameters whatsoever) is consistent but has no realization.

In general, it's often helpful to find some "finiteness" property which elements of the structure have; such finiteness properties probably aren't first-order expressible, and so indicate how to find failures of saturation.