Why do we need vectors and who invented it?
Vectors should be thought of, at a first approximation, as "numbers with direction". For physical phenomena which carries a direction, such as velocity and displacement, vectors are immensely useful.
The concept of a number with direction most likely dates to antiquity, as the making of maps and sign-posts already implicitly incorporates the notion. The modern representation of a vector/point in space with an ordered triplet of numbers is often attributed to the advent of analytical geometry due to the philosopher Rene Descartes.
A different notion of vectors also arose with the "discovery" of the complex numbers by Jerome Cardan: the imaginary numbers can be thought of as living on a different direction as the real scalars (so the complex numbers form a real vector space).
Over the past 400 years or so the notion of vector gradually evolved to become what we know today, with contributions from branches of mathematics that developed into modern analysis and algebra. A nice summary of that period of development is available here. See Michael Crowe's book for a fuller description of also the Greek contributions and the influences from the 16th century in this matter.
In short, vectors shouldn't be thought of as being "invented", nor should it be attributed to one person alone.
As with most anything in physics, the addition of forces follows the parallelogram law exactly because it agrees with every experiment mankind has ever made---in other words, ultimately it's an assumption we make based on what we see. But, it's perfectly intuitive that this is how forces should behave: say you're at the origin on the xy-plane and have a force pushing you in the positive x-direction and one pushing you in the positive y-direction, both of equal magnitude. Then it should physically and intuitively clear that the net result will be a force pushing you in the direction of (1,1). If the force in the x-direction was stronger, then the net result would still push you in some "diagonal" direction, but aligned more with the x-axis. This is exactly the parallelogram law.
Same exact thing. The problem is that we can't really picture what a right triangle in 4 dimensions looks like, but it's exactly the same idea.
Of course, there are many possible "dot products" that can be defined on vectors, and in many applications it is indeed useful to use one different from the standard one. But I would say the importance of the standard dot product indeed comes from its relation with the cosine of the angle between them: again if you try to think about this physically, this is a precise measure of how much the two vectors (or forces if you like) are working "with" or "against" each other. Note in particular that the sign of the dot product solely depends on whether or not the angle between your two vectors is less than, equal to, or greater than 90 degrees.
Willie covered a lot of what I wanted to say; however I'd like to make a little historical digression: before we ever had the concept of a vector, there was the quaternion, William Rowan Hamilton's generalization of the usual complex numbers. They proved very convenient for physical applications, and thus the use of quaternions took off. In fact the electromagnetic equations of Maxwell were first couched in quaternion notation.
It was Josiah Gibbs and independently Oliver Heaviside who looked at decomposing the quaternion into a scalar (real) part, and a vector (imaginary) part, and found that the manipulations in this new formulation were "cleaner". Vector analysis took off, and quaternions became less prominent.
I'd say more, but a book has already been written about this matter, so I refer you to it.