Hedge is a term used in trading that simply means opening a position in order to reduce risk. This is usually done by either buying an asset that moves in the opposite direction to your main position, or by taking the opposite position in a correlated asset (such as a derivative).
For example, if the portfolio of an investor is fairly correlated to the health of the economy, they may take out positions in gold (which usually rises if the economy is doing badly) to hedge their risk.
Hedging is used to reduce risk and is typically done through the use of derivatives such as options and futures, although it may be done using other assets (i.e. gold, government bonds).
A perfect hedge will mean there is zero risk in holding a position (or a combination of positions) although this is extremely hard to obtain as very few assets are perfectly correlated or move exactly the same amounts.