Fossil fuels are a category of fuels that are made by geological processes acting on dead organisms, often hundreds of millions of years old (For reference, please see the carboniferous period and then click on 'prehistory' to get some perspective on how long ago this was). Fossil fuels are not considered a renewable energy source because they cannot be reproduced at the rate of which we are consuming them. Whether or not they should be considered a sustainable energy source is a difficult question to answer. According David MacKay, the Chief Scientific Advisor to the UK Department of Energy and Climate Change from 2009 to 2014, this answer is no if we follow the "business as usual" fossil fuel consumption rates.  Fossil fuels include: coal, oil, and natural gas; and can include peat. Chemically these fuels are mainly composed of carbon and hydrogen with some oxygen, nitrogen, sulfur and a host of other smaller elements. All of the energy in fossil fuels initially comes from the sun, so fossil fuels are a long term store of solar power. These dense supplies of energy provided 87 percent of global primary energy consumption in 2012.  We use fossil fuels for everything from generation of electricity to home heating to making transportation fuels.
The burning of fossil fuels results in the release of carbon and hydrogen compounds that combine with oxygen from the atmosphere to form carbon dioxide and water vapour in a process called combustion. In addition to greenhouse gasses, fossil fuels cause other pollution as well, including but not limited too NOx, SOx, particulate matter (PM), volatile organic compounds (VOCs), carbon monoxide (CO) and mercury. There have been tremendous improvements in the environmental impacts of fossil fuels with more stringent pollution control measures being put in place in response to legislation. By far the biggest concern about using fossil fuels is the impact on the climate. The large amount of carbon dioxide released by burning fossil fuels is causing changes to the Earth's climate. There are a number of other environmental concerns beyond the burning of these fuels especially during their excavation-habitat destruction, biodiversity loss, and water pollution - to name a few.