Balsamic means 'like balsam’ - and balsam is an aromatic resin - balsamic vinegar simply refers to the fact that it is thick (resin like) and aromatic.
Here is a short summary of Balsamic Vinegar, where it comes from and how it is made.
The unique and traditional balsamic vinegar of Modena, Italy is made from the 'must' (unfermented juice) of mainly the Trebbiano grape, other grapes used are Lambrusco, Ancellotta, Sauvignon and Sgavetta. These 'musts' cannot have anything added. The must is then boiled down in open pots over a direct flame.
The extract (concentrated juice) from this cooking is now a fruity syrup. At this point some 'mother' of vinegar can be added. ('Mother' is a stringy, slimy substance that forms on the surface of vinegar, composed of various yeast and bacteria [especially mycoderma aceti] that cause fermentation in wine and cider, and turn it into acetic acid - vinegar). It is then aged in barrels of different woods - first in one, then transferred to another etc.
Each company has its own secret progression of wooden barrels usually including chestnut, ash tree, cherry, mulberry, juniper and oak. The finished vinegar must be at least 12 years old, and some is aged much longer. The finished vinegar is then presented to the DOC, a governing body similar to those that govern the quality of French and Italian wines. Balsamic vinegars without this designation on the label are usually unaged, aged for 6 months to a year in stainless steel tanks, or aged for 2 to 12 years in wooden barrels.