"Aggressive" refers to the bicycle frame's geometry. Specifically, the angles of the head tube and seat tube to the horizontal. The more upright these angles, the more "aggressive" the frame is said to be. The effect of upright angles is to make the bike steer more quickly and to reduce the effects of the wheels' gyroscopic inertia. A bike with aggressive angles will be called a "twitchy" steerer. Such angles and steering characteristics are prized among racers, who must often make radical steering corrections to avoid spilled riders ahead.
In past decades (specifically, the 70's), racing bicycles typically had 72.5 degree head tube and seat tube angles. "Touring" bikes with less aggressive geometry typically had anywhere from 69 to 72 degree head and seat tube angles.
Today, it is not unusual to find "sport" bicycles with 73, 74, and even 75 degree head and seat tube angles. Such bikes are extremely "fast" and "twitchy" in the steering department. They are profoundly unsuitable for non-racing riders, despite being sold universally to riders who never plan to race.
Less aggressive geometry imparts stability to the ride, allows the rider to cycle securely even with no hands on the bars, and is more "forgiving" of unexpected potholes and stones in the road. You pays your money and takes your choice...