Origin
The origin is the point (0, 0) where the x-axis and y-axis intersect on the coordinate plane.
Definition
The origin is the point $(0, 0)$ where the x-axis and y-axis cross, the "address" from which every other point is measured by distance and direction. It is the additive identity of the vector space $\mathbb{R}^2$, the zero vector $0$ and identity element of the additive group $(\mathbb{R}^2, +)$: any vector added to it is unchanged, and it is the fixed point of all dilations centered there, so a dilation with center at the origin simply multiplies coordinates by the scale factor, $(x, y) \mapsto (kx, ky)$. In projective geometry the origin loses its special status, since all points become equivalent under projective transformations, and in complex analysis $z = 0$ is often a critical point or singularity of functions like $1/z$.
Example
If you draw a coordinate grid on paper, the origin is the point right in the middle where the two lines meet, and every other point is described by its distance and direction from there. Rotating point $(3, 0)$ by $90$ degrees counterclockwise about the origin gives $(0, 3)$, and scaling from the origin by factor $2$ maps $(3, 4)$ to $(6, 8)$. A linear map $T: \mathbb{R}^2 \to \mathbb{R}^2$ always satisfies $T(0) = 0$: linear maps always fix the origin, while affine maps (linear maps plus a translation) do not necessarily fix it.
Key Insight
The origin is like the address $(0, 0)$ on a map: everything else is described by its distance and direction from this center point. The distinction between linear and affine transformations, whether the origin is fixed, is fundamental in graphics, robotics, and physics, since changing the origin (a translation) can simplify problems by exploiting symmetry.