In physics, what is defined as the product of an object's mass and velocity?

The product of mass and velocity is momentum.  

A good way to summarize momentum is to say momentum is mass in motion. All objects have mass, but not all objects are moving; therefore, not all objects have momentum. If an object is moving, though, it has momentum. Because the equation for momentum involves multiplication, increases in either mass or velocity will result in the increase of momentum. That means a football linebacker can increase his hitting power by gaining mass, becoming faster, or both. 

Some students occasionally confuse momentum and inertia. All objects always have inertia because inertia depends only on mass. Inertia is a resistance to changes in any motion. An object will have the exact same inertia whether it is moving or not, but an object will have zero momentum if it is stationary. That particular object's momentum will increase as it begins to increase its velocity.