Hypothesis and model class

The hypothesis class (or model class) $\mathcal{H}$ is the set of functions the learning algorithm is allowed to choose from. Linear regression uses $\mathcal{H}=\{f(\mathbf{x})={\mathbf{w}}^{\top }\mathbf{x}+b\}$; a decision tree of depth 3 uses all trees with at most 3 levels; a neural network uses every function its architecture can represent.

Choosing $\mathcal{H}$ is choosing your inductive bias — the assumptions that let you generalise from finite data. Without any restriction, infinitely many functions fit the training data perfectly and disagree everywhere else.

A small class may not contain anything close to the truth (approximation error → underfitting). A huge class contains many functions that fit the noise (estimation error → overfitting).