Simply: Filtration is the process of removing suspended particulates from wine resulting from the fermentation process. And very important for future clarity and stability of a wine.
A process of 'cleaning up' a wine - used after fermentation and ageing, and before bottling; it is similar to running coffee through a filter, but arguably not always necessary to produce fine wine. The purpose of filtering is to remove sediment, grape skins, dead yeast, crystals and bacteria that can adversely affect wine quality.
The degree of filtering can range from very fine to coarse; however, it is increasingly being minimized (or avoided whenever possible) because the finer the filtering, the more flavours and characters (e.g. palate texture) are also removed from the wine.

 

Today - many wineries (old and modern) are using the more labour-intensive, old-traditional practices of fining or racking to clarify their wines. Up until the late 1970's many filters were made from asbestos material.
Filtration is also used to ensure clarity, plus physical stabilisation prevents the formation of haze and deposits after bottling, while microbiological stabilisation eliminates yeasts and bacteria that can demage a wine's taste. Careful use of precise filtering pads and agents allow the winemaker to target specific suspended substances based on their size.
Some winemakers feel filtering reduces the quality of wine. Emile Peynaud - distingished enologist at the University of Bordeaux had this to say about the filtering debate:
"Resistance to the practice of filtering arose from the reproach made that it tended to thin down and emaciate the wines. Nevertheless, if every precaution is taken - it may be stated that the mechanical action of filtering has never had a negative influence on quality. To suggest the contrary would mean conceding that the foreign substances - which filtration is precisely designed to remove, have a favourable influence on taste".
Filtration is also a key stage in the elimination of the deposits formed in sparkling wines during its secondary fermentation inside the bottle.