What is the dot product and why do we need it?

Dot products are very geometrical objects. They actually encode relative information about vectors, specifically they tell us "how much" one vector is in the direction of another. Particularly, the dot product can tell us if two vectors are (anti)parallel or if they are perpendicular.

We have the formula $\vec{a}\cdot\vec{b} = \lVert \vec{a}\rVert\lVert \vec{b}\rVert\cos(\theta)$, where $\theta$ is the angle between the two vectors in the plane that they make. If they are perpendicular, $\theta = 90^{\circ}, 270^{\circ}$ so that $\cos(\theta) = 0$. This tells us that the dot product is zero. This reasoning works in the opposite direction: if the dot product is zero, the vectors are perpendicular.

This gives us a quick way to tell if two vectors are perpendicular. It also gives easy ways to do projections and the like.

That's a huge question. It's what's called an inner product. A good short answer is that it gives you a way to make sense of what an angle between two vectors is. $$\theta = \cos^{-1}\left( \frac{a\cdot b }{|a||b|} \right)$$