My book says that the skew of an implied volatility smile is the slope of the curve.
But then it says that the skew is 0.25 %, i.e., if K goes DOWN 1, implied volatility goes UP 0.25 %.
However, isn't the skew then -0.25%?
No - Volatility skew is the change in volatility when moving away from at-the-money, and is almost always positive (by convention, not by definition). I'm assuming that the example in your book is K moving down from at-the-money (closest strike to the current price) for call options, which means the strike is getting deeper in-the-money, and the volatility is going up by 0.25%.
Remember that there is both a "call" skew and a "put" skew - one when the strike goes down (call) and one when the strike goes up (put). If you think of it as "slope" then it would make sense for the call skew to be negative, but the convention is to measure the increase in vol as you move away from at-the-money, so the "call skew" is the increase in vol per unit decrease in strike.