Fungi are plants that feed on other plants. They reproduce by means of 'spoors', small (microscopic) seeds which are distributed by the wind, or water. Should a spoor be able to enter your tree, perhaps through a cut or other wound, it will begin the process of breaking down the tissue, killing the tree. Modern fungicides will get rid of an established fungus, but often, by the time you notice the fungus, the damage is done. They are best used to prevent fungi from becoming established, or to control their spread. Fungi are quite common in all kinds of gardens and especially with fruit trees, which need to be sprayed routinely.
'Mildew' and 'Black spot', are two of the most common fungi you will come across and unlike their mushroom producing cousins do not invade and eat the tree. Mildew coats leaves with a pale blue/grey powder, while blackspot (beloved of rose growers) peppers small and sometimes not so small black patches over the leaves. Both reduce the leaves ability to produce energy for the tree and may, in extreme cases kill the tree.
Both are readily removed with commonly available fungicides, however the leaves will never return to a good condition and are best removed. My own preference is for the 'Fixed Copper' fungicides, commonly known as , Bordeaux mixture, Bordeaux Mix, copper dust or liquid copper. Care should be taken not to get them in the leaves, particularly of deciduous bonsai, as they will leave a powdery residue which is difficult to remove, but will not actually harm the tree.
They are usually sold as a concentrate in wettable powder form. Basically is's a mixture of copper sulphate and hydrated lime which gets its name from years of use in the French vineyards. As the mixture contains only 'natural' ingredients, It may be considered organic.