Those questions are somewhat vague; how much detail are you looking for?
Fungi grow the same way all living things do: by taking in nutrients, processing them, excreting waste products, cellular division, tissue differentiation (for the multicellular species), etc. When you see mushrooms in the forest, they are not actually the fungus itself; they are only the reproductive organ of the fungus. The main fungal body lives underground and is composed of a series of thread-like filaments called the mycelium. What's interesting is that fungi have holes in their cell walls which allows cytoplasm to flow efficiently from one cell to another. This allows fungi to grow quickly (ever seen mushrooms sprout up overnight?).
Fungi cannot create their own energy from photosynthesis, so they absorb nutrients and sugars from other organisms (often plants). They may not even need to parasitize living organisms, but dead ones instead (mold on bread being an example).
Fungi are composed chemically of largely the same basic substances as other living things: water, nucleic acids, carbohydrates, amino acids, trace minerals, etc. Their cell walls contain a material called chitin, which is similar in some ways to cellulose in plants, but contains nitrogen. Many fungi (in the form of fruiting bodies, i.e. mushrooms) may also contain toxins.