With stock options, what is the difference between a high "delta" option and a low "delta" option?

Sensitivity to the movement of the underlying stock price. As the previous poster noted delta is a measure of how much the option price moves relative to the underlying stock price. In call options the delta is positive to the price of the option, so a delta of 0.6 would mean for every $1 move in the stock price the option price would change 60 cents. The opposite is true with puts which would be effected in the opposite direction. So the higher the option delta, the greater sensitivity to option has to the stock price. Options that are in-the-money and close to expiration date will almost always have higher deltas than those that are not. Delta will also increase the closer the option contract is to the expiration date. American option contracts (can be exercised at any time) will have higher delta than European options (can only be exercised at expiration) .

Delta is basically change in the value of an option(call/put) for a unit change in the value of the stock. For example if the stock increases $10 causing the call option to increase by $6, then the delta is 0.6 ...

There is no fixed range for high or low delta and depends from stock to stock.