There are two velocities for each collision, corresponding to two objects: object 1 and object 2. (they should have used A and B, it would have been clearer)
The far left column are the conditions of the collision
The top of the column is (obviously) the type of collision.
The equations simply tell you the relationship of the velocity of the object to the masses of the two objects, and how that varies with each type of collision.
Take V1 for an Elastic collision.
The equation tells us how the velocity of object 1 is related to the mass of the objects 1 and 2 and to the initial velocity, Vo. Here it says that the ratio of the velocity of object 1 to the initial velocity is equal to the difference between the two masses involved in the collision, divided by the total mass.
Now, if the mass of object 1 (m1) is less than the mass of object 2 (m2) [m1<m2] the velocity of object 1 after the collision will be greater than -Vo (or less than Vo) and less than 0. The reason it is put this way is because the direction of object 1 will change after the collision; it will reverse. This makes sense intuitively if you think about it--think of a marble (m1) striking a bowling ball (m2); the marble will bounce back and reverse direction, but with less velocity than it had initially. Because velocity is a vector, we know that the negative velocity means in the opposite direction. The next condition (m1=m2) tells us that if the two masses are equal they will, in an elastic collision, collide and then simply stop. The last condition tells us that if the mass of object 1 is greater than the mass of object 2 (m1>m2) the velocity of object 1 will be in the same direction as the initial velocity (Vo) but will be less than the initial velocity (because of the collision).
The other formulae and conditionals all express the same relationships but with different types of collisions.
I hope I helped a little, if I even was looking at the same chart. Good luck!
