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Delta hedging is the process of setting or keeping the delta of a portfolio of financial instruments zero, or as close to zero as possible - where delta is the sensitivity of the value of a derivative to changes in the price of its underlying instrument. Mathematically, delta is the partial derivative of the portfolio's fair value with respect to the price of the underlying security; see The Greeks. Being delta neutral (or, instantaneously delta-hedged) means that the instantaneous change in value of the portfolio for an infinitesimal change in the value of the underlying is zero.
Keeping delta at zero is termed a "static delta hedge"; keeping delta close to zero is a "dynamic delta hedge". Delta constantly changes, thus, once the delta of a portfolio has been made zero by adjusting its holdings (typically in the underlying security for a portfolio of derivatives) it is zero only for that instant; delta neutrality is instantaneous.
The term static delta hedge is therefore a misnomer and thus (re)setting delta to zero is often preferred. In dynamic delta hedging, the portfolio is readjusted regularly in order to reset the delta to zero. Between readjustments, the portfolio delta will deviate from zero.
As above, a portfolio has to be adjusted continuously (i.e. infinitely often in any time interval) in order to maintain absolute delta neutrality. This idea plays an important part in the Black-Scholes model of option pricing, and - indeed - the expected cost of keeping a position in one option, the underlying asset (and cash) delta neutral is equal to the initial fair value (Black-Scholes price) of the option; for the underlying logic see the discussion at Rational pricing.
In fact, the amount by which a hedge has to be adjusted to stay delta neutral is related to gamma, the second derivative of the portfolio value with respect to the price of the asset in question. For example, if a position is 'long gamma', i.e., has a positive gamma, an increase in the asset price will lead to a positive delta, and one will need to sell some of the asset to 'flatten' the delta. Similar, a decrease in asset price will cause one to buy more of the asset. From this it is intuitively clear that a high volatility of the underlying asset will lead to trading profits.
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