Two vectors, and the answer is a plain number with no direction at all — which is why it is also called the scalar product. From components: $\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y$, read aloud a dot b equals a-x b-x plus a-y b-y, where $a_x, a_y$ are the components of the first arrow and $b_x, b_y$ those of the second — the subscript names the axis, the letter names the arrow. Like with like, then add.

From lengths and the angle between them: $\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta$. Two doors, one room — and setting them equal is how the next lesson finds an angle. What the number MEANS is the cosine: it is the two lengths multiplied, then taxed by how far apart they point. Same direction, $\cos 0^\circ = 1$, full value. Perpendicular, $\cos 90^\circ = 0$, and the dot product is exactly zero — the cleanest perpendicularity test in mathematics, and one you will reach for constantly. Past $90^\circ$ it goes negative, meaning the two broadly disagree. A negative dot product is information, not a mistake.