Scope of this answer
As with any machine or useful artefact, the structures (including shape and surface chemo-electrical properties) of proteins determine their ability to fulfil the functions they perform. This relationship (and the way in which the structure arises from the amino acid composition and sequence) is illustrated in any textbook on biochemistry, which typically devote whole chapters to individual proteins. That, paradoxically, makes it difficult for the naïve student to obtain an overall perspective of the topic before going into more detail. This is the purpose of my answer, which is illustrative rather than exhaustive, and merely provides links to detailed accounts of different proteins. I shall use Berg et al. Biochemistry at NCBI Bookshelf for all my examples.
I shall not deal with prediction of function from structure. That is covered in another answer.
It seems to me best to start with functions and then indicate the general structural aspects, followed by detailed structural features that can accomplish such functions.
1. Acting as fibres
The spider’s web, the maiden’s flowing locks, the athlete’s muscles all constitute and depend on the properties of fibres which are protein in nature. All share the shape of having an extended chain. Each differs in how the chains are constituted, and with what other proteins it might interact (e.g. to give muscle its contractile properties).
Berg et al. include Fibrinogen as an example of fibre formation.
2. Binding of small molecules in an aqueous environment
Many proteins have a function that involves binding small molecules in an aqueous cellular environment. To do this they need to be folded into a roughly globular shape with a surface that renders them soluble in water, and they need to have a specific binding site for the molecule — an indentation in the surface of the correct shape and electrical charge. Examples of such proteins are those that bind oxygen (e.g. myoglobin and haemoglobin), antigens (immunoglobulins) and steroid hormones (soluble receptors). These proteins differ in which molecules they can bind (the binding site is specific) and differ in what events follow (are triggered or are possible) following binding. These latter are determined in a more or less obvious way by other aspects of the proteins structure. For example in the case of immunoglobulins which have two arms, the possibility of cross-linking to form larger aggregates occurs.
Berg et al. have a short introductory section on myoglobin and extensive treatment of haemoglobin, which shows how its subunit structure allows its function to be fine-tuned.
3. Binding of small molecules at the membrane
This is analogous to 2, above, but here the overall structure is partly hydrophobic, so that part of the protein can interact with and be embedded in the membrane. In this case the binding of a molecule (hormone, agonist) generally causes a change in shape, allowing interactions with other molecules or catalytic function, transmitting a signal across the membrane.
Berg et al. have quite an extensive section on an abundant class of membrane receptors.
4. Transporting molecules across membranes
Cell membranes are impermeable to polar molecules and special channels are needed to allow their passage. For a protein to be able to act as a membrane channel it needs to fold so that it has hydrophobic exterior interacting with the membrane, and (simplistically) a polar ‘hole’ through the middle, tailored to the specific molecule it will transport. The types of channel vary, but there are a set of common designs, some of which (such as porin channels) are more easily recognized visually than others (e.g. calcium ion channels).
Berg et al. consider the system involved in pumping calcium across membranes.
The largest category of proteins is enzymes — the proteins that catalyse the chemical reactions that allow life to occur. Although these are generally globular in shape, they have specific binding sites for the reacting molecules (substrates) — like the binding proteins. However, in addition, there are generally strategically positioned specific amino acid side-chains that actually participate in the reaction, often providing a pathway with a lower energy barrier. Families of enzymes exist with such similar reaction mechanisms, but working on different molecules (e.g. the digestive enzymes trypsin and chymotrypsin).
Berg et al. section 9.1 considers structure–function aspects of serine proteases such as trypsin.