To be honest, there are far too many proteins with varying structures and functions to classify in such a general way. I'll give two examples of protein structure and function based on two canonical types of proteins: membrane potassium channels (see first link) and enzymes (see second link).
Potassium channels are highly selective membrane proteins that only allow potassium to diffuse down its concentration gradient. The structure of the protein correlates highly with its function in two primary ways. The first way is based on the face that the channel is a membrane protein. The fact that it is a channel means that the protein must have residues that interact with the hydrophobic lipid bilayer and residues that interact with the aqueous intracellular and extracellular environments. This means that the residues that interact with the lipid bilayer will be hydrophobic and that the residues that interact with the inside or outside of the cell are hydrophilic. The structure of the inside of the channel where potassium ions pass is just as important. Negatively-charged amino acid residues at the recepticle of the channel interact with the positively-charged potassium ion, but the residues are a certain distance away that would not allow sodium or other ions into the channel. By having a specific structure, the potassium channel achieves its function with very little error.
Enzymes are another case in which structure interacts intimitely with function. Enzymes fold into specific structures based on the hydrophobic and hydrophilic amino acids in their primary structure. When the enzymes fold in a specific way, they have a site, which we call the "active site." In one of two primary models of enzymes, the active site has a structure that mimics the transition state of the exact reaction it is meant to catalyze with other residues to hold the reactants in place. In the other model, the active site resembles a "lock" which a reactant ("key") and only the specific reactant can fit into. In this way, the enzyme marries structure and function for its purpose in the body.
These are only two examples of the relation between protein structure and function. There are plenty of others, and I encourage you to search for them! Good luck!