The basket press is the traditional way of extracting juice or wine from the grape skins, and is more gentle on the grapes than newer designs.
Like it sounds, a basket press is essentially a big basket into which the crushed grapes are loaded. A flat plate is lowered down on top of the grapes and presses the liquid out through small holes in the side of the basket. Traditionally the basket is made from wood, but stainless steel is a more modern (and easier to clean) option which does exactly the same thing.
Basket presses are quite labour intensive, and most wineries now use the newer Airbag or tank presses which are much easier to use. These airbag presses rotate and break up the "cake" of skins to extract more liquid, and this tends to result in harsher tannins in the wines.
Basket presses are generally associated with premium red wines such as Shiraz. We use our basket press for all our wines - red or white. It is the only press we have, and the only press we want!
For reds it is great because it is more gentle and we get softer tannins in the wines (all the tannin is in the skins, and is the drying character in red wines). For whites it is great because there is a large draining area for the juice to get out really quickly, and the less time white juice is in contact with skins the better, because we don't want the tannins.