When it's said a piano has crisp "attack", what does that mean?

Think of attack as how quickly the sound begins once you've struck a key. In pianos the attack is sudden; no sound and then BANG! there it is. Contrast this with the pipe organ, particularly notes played on the great bass pipes of the pedal. You will hear another sound for a tiny fraction of a second as the note begins. The pipe sort of 'warms up' before it speaks; before the moving air produces its vibration. This is called 'chiff' (the trivia word for the day).

A piano with a crisp attack is very responsive. It plays fast, aggressive pieces well.

Every note of music has three phases: Attack, Sustain and Delay. The 'attack' simply signifies the very start of the note and, in some cases, the way that note is caused to start sounding. Being a percussion instrument (with strings being struck by hammers), a piano will very often have an abrupt, 'crisp' attack. More graduated attacks can be created on other instruments through methods such as breath control (on wind instruments) and careful use of the bow (on string instruments).