Animals (multicellular life) evolved about 600 million years ago, during the Ediacaran period. Life in general existed much earlier than that, at least 3.8 billion years ago, and possibly as far as 4.2 billion years ago, just 300 million years after the formation of the Earth itself. Thus, about seven-eighths of the history of life occurred before animals came onto the scene. The period of evolution before animals spans at least 3 billion years.
Evolution before animals is thought to have been relatively slow. Some organism similar to modern-day cyanobacteria appeared at the beginning, taking up the Earth's carbon and forming it into photosynthesizing biomass. This is evidenced by isotopic studies of various rocks -- since life preferentially intakes the lighter isotope of carbon, high abundance of this light isotope in rock is considered indicative of life. Diamonds dated to 4.2 billion years ago found in Australian zircons have had this isotope ratio, leading some scientists to argue that simple life may have existed that early.
Evolution before animals took place almost exclusively in the oceans. Around 2.7 billion years ago, life developed oxyphotosynthesis, which pumped large quantities of oxygen into the atmosphere, which at the time was mostly carbon dioxide and water vapor. This event, called the "Oxygen Catastrophe," lasted for over 500 million years, and when it was done, the Earth had a percentage of oxygen roughly similar to today's. This is called a "catastrophe" because it would have killed many organisms to which oxygen was poisonous. This is one of the most significant events in the history of evolution before animals: the transformation of the Earth's atmosphere from an anoxic to an oxic state.
Another important event in evolution before animals was the evolution of eukaryotic, or complex cells. A eukaryote is any cell more complex than a bacterial cell. Eukaryotic cells are larger than bacterial cells, about 10 times larger, and contain organelles, or specialized parts. Some organelles, such as mitochondria, the "power plants" of the cell, are likely bacterial cells that were captured, retained, and replicated in future eukaryotic cells. Eukaryotic cells evolved between 1.6 and 2.1 billion years ago. The first eukaryotic cells were likely red algae.
Another significant event prior to the evolution of animals were the Cryogenian glaciations -- which occurred between 790 and 630 million years ago. During this time period, glaciation was so severe that there is evidence of thick ice existing at the equator. Some scientists even think that the oceans may have frozen over -- a scenario known as "Snowball Earth." Whether or not the oceans froze, such a cold environment would have certainly been hostile to multicellular life, and it is interesting that it appears to have emerged relatively quickly after the conclusion of this glacial period.