Spores are the main dispersal form to spread a fungus beyond its immediate environment. These are sometimes produced asexually, in non-specialized structures, but to make fruiting bodies, two genetically-different individual hyphae must come in contact. The cells in hyphae are haploid, with only one set of chromosomes. The hyphae fuse and share cell contents, including nuclei. The nuclei eventually fuse into diploid nuclei, and inside the fruiting bodies, spores are produced by meiosis; depending upon the total number of chromosomes, this generates variety in the fungi that the spores will germinate into. Because the new fungi are genetically distinct from the "parents," this is a form of sexual reproduction.
If spores are produced inside a special cell, the asci, this is the basis of the major subgroup Ascomycetes; if spores develop on projections of special basidia cells, it's the subgroup Basidiomycetes.
Spores may just be released into the air, but some fungi have special "launching" processes to get the spores clear of the potentially-sticky surfaces of the fruiting body. In most mushroom, the spore has a wet coating almost all of the way around; in a dry area near where it attaches, a small blob of sugar is produced. Sugar molecules have a attraction for water molecules, and a drop of water forms and grows there, bending the support structure down. The drop eventually gets big enough to contact the wet surface of the spore, which makes the drop release; the spring-back of the support tosses the spore free of the "fins" inside the mushroom cap, so it can drop freely into the air below and get carried away.
Yeasts are a special-case fungus: they are single-celled, so do not form hyphae, and they reproduce by budding, a reproductive process where the offspring are much smaller than the original. Slime molds are sometimes thought of as fungi but are actually protozoans, a type of ameba with some odd abilities: in low-nutrient conditions, they can fuse together as a larger (and more mobile) "giant ameba"; in very bad conditions, they assemble into spore-producing fruiting bodies, which is why they are sometimes thought of as fungi, although their genetics say otherwise.