Microorganisms are microscopic organisms that exist either as single cells, which are invisible to the naked eye, or as cell clusters. In great enough numbers, cell clusters become visible without the aid of a microscope. For example, to be able to see a typical group of bacteria there need to be approximately a million or more cells.
All living organisms can be broadly divided into two groups: the Prokaryotes, and the Eukaryotes. The prokaryotes are microorganisms, while the eukaryotes are multi-cellular organisms, that is, animals and plants. One of the main differences between these two groups is that eukaryotic cells contain a membrane bound nucleus, and other other membrane bound organelles (which are like the 'organs' of the cells, making energy etc), while the prokaryotes do not. Also, prokaryotes process cell walls which are stronger than the mebrane (the phospholipid bilayer) which surrounds eukaryotic cells. It is this wall which allows prokaryotes to survive as single cells.
The prokaryotes - microorganisms - can again be divided into two groups: the Archaea, and the Bacteria. In the evolutionary line, the archaea are much older than bacteria, and are believed to be the oldest form of life on Earth. There is an immense amount of research into microorganisms (this study is known as microbiology) and so far there have been literally thousands of identified species of microorganisms.
Not all microorganisms are microscopic. For example, fungus is classified as a microorganism, because it is formed of microscopic components, and can exist independently as such. However, fungus can and frequently does form clusters that become clearly visible to the naked eye. Mushrooms are a good example of this. In fact, the largest known organism on record was actually a microorganism. Researchers discovered a single colony of the fungus Armillaria bulbosa which was found to occupy 15 hectares of forest, and in total was calculated to weigh approximately 10,000kg. To reach this size it must have been growing for 1,500 years or more.
Other than archaea and bacteria, the only other type of microorganism is the virus. The virus is not cellular, which is why it is not classified above. However, they are indeed microscopic, often being smaller than most bacteria, and add to the diverse range of microorganisms which surround us.
NB: There is continuing debate as to whether or not virii are actually life forms, as they fail to fulfil several of the criteria, such as excretion.