Why and how does negative velocity exist?

I think one of the main reasons that you have velocity is to isolate a particular direction of movement from your forward speed.

If you travel North north east, you can extract the speed at which you move eastwards by calculating your eastwards velocity (possibly 1/3 of your speed travelling NNE).

Negative velocities probably arrived as a consequence of the fact that when measuring a velocity, you have to define a direction.

From the math point of view, you cannot have “negative velocity” in itself, only “negative velocity in a given direction”.

The velocity is a 3-dimension vector, there is no such thing as a positive or negative 3D vector.

However, if you consider the velocity in direction $mathrm{x}$, where $hat{mathbf{e}}_{mathrm{x}}$ is some unit vector giving a reference direction (say, "West"), then the velocity “in direction $mathrm{x}$” is simply the scalar product of the velocity and $hat{mathbf{e}}_{mathrm{x}}$. This quantity is a real number and can be negative. If it is negative, it is equal to $-1 times text{(velocity in direction -x)}$: compute the velocity in the opposite direction, and reverse the sign.

I will only consider one dimensional motion(motion along a single axis).The main objective of terms like position and velocity is to describe the motion of an object easily.

We define velocity to be the rate of change of position .By convention we choose a fixed point (along the axis of motion) and call it origin and define an object's position on that line based on the distance from this point. Again general convention is that distances measured towards right are positive and those measured towards left are negative (you can use them reversed if you want).You can easily see that in defining measurement of position this way we have covered all the points on the axis.In these conventions a position 2m means 2m right of origin and a position -6m means 6m left of origin.

Now you would have heard that the sign of velocity gives direction, but first of all directions are again just references made by us .By trial and error we can make out the physical meaning of + or - sign . (By convention you call the direction in which the negative numbers increase negative direction(i.e left in this case) and the direction in which positive number increase the positive direction (i.e right in this case)).

Example:-

You see an object first at 2m and then at -3m after 5s . You say that it has moved towards left or in the negative direction . Now by the definition of velocity you calculate change in position (-3-2)m = -5m (analysing you can see that the "-" came out automatically as a result of our convention) and change in time = 5s. Dividing you get velocity as (-1m/s) or you can say that the object was moving towards left at 1m/s .In this way you can figure out what the negative sign means . It simply means that the motion is towards the negative direction or towards left.(As you can see our definitions and conventions help us describe completely the motion of an object , both the rate at which it moves and where it moves)

In the computer language FORTH the turtle can go anywhere in the plane, going always 'ahead' X units and turning left or right by a Y angle units. The turtle ignores the notion of negative and yet it moves i.e. in any referential the space coordinates vary in time , and it has a velocity.

The Question and several of the Answers do not know the difference between the positive nature of any amount of a physical quantity and the representation in a referential that was constructed by convention and ease of use. Compare the Polar referential there are no negatives and the usual Cartesian one.

By example : the length of something, the distance from here to there is always positive.

A vector is a pair of magnitude and direction, the length of it is positive by definition.

When we represent that an object move from the position 0 to -X and then back to 0 we can not say that it moved 0 units (-4+4=0) of length . In fact it moved twice the length 4, i.e. 8 units of length.
If it took 2 seconds in that motion (see velocity versus speed notion) than we can not add the two vectors and say the .. is 0.
That link provides a distinction between two different notions of speed in the textbooks and the concept of velocity.

def 1 : $s=frac{distance}{Delta t}$ (it depends on the path)
def 2 : $s=left|frac{vec{v}}{Delta t}right|$

Negative and positive is arbitrary. If I defined north as positive, south would be negative. If I defined south as positive, north would be negative. The signage merely serves to provide direction for the velocity vector relative to some defined positive direction. All directions are arbitrary, and you can create any coordinate system for your events as long as everything is consistent with eachother.

We need this convention to explain position as a function of time, for example. If velocity was fixed to be positive, or similarly, if it were scalar, mechanics would have some issues, since it would mean an object could never decelerate let alone go backwards.