You can define the projective plane $P^2$ as the space of lines in $\mathbb{R}^3$ through the origin. In particular, given a line in $\mathbb{R}^3$ which passes through $(0, 0, 0)$, there is a corresponding point in the projective plane $P^2$.
Suppose $\ell$ is a line in $\mathbb{R}^3$ through the origin. Any line is uniquely determined by two different points on the line. We know that $(0, 0, 0)$ is a point on the line, so suppose $(a, b, c) \neq (0, 0, 0)$ is a different point on $\ell$. The point of $P^2$ corresponding to the line $\ell$ in $\mathbb{R}^3$ is denoted by $[a, b, c]$. Note that we could have chosen a different point on the line $\ell$, but every other point is of the form $(ka, kb, kc)$ for some non-zero real number $k$, so we need $[a, b, c]$, the point in $P^2$ representing $\ell$, to be the same as $[ka, kb, kc]$ because it represents the same line. That is, we require $[a, b, c] = [ka, kb, kc]$ for all non-zero real numbers $k$. This is why we use homogeneous coordinates on $P^2$.
With this in mind, we can now understand the relationship between points in $\mathbb{R}^3$, other than the origin, and points in $P^2$. Given any point $(a, b, c) \neq (0, 0, 0)$ in $\mathbb{R}^3$, there is a corresponding point $[a, b, c]$ in $P^2$ which represents the line in $\mathbb{R}^3$ which passes through $(0, 0, 0)$ and $(a, b, c)$. The reason this correspondence fails for $(a, b, c) = (0, 0, 0)$ is that you don't have two different points in $\mathbb{R}^3$, so you don't get a line (and hence, you don't get a point in $P^2$).
Note that a point at infinity of $P^2$ is a point of the form $[a, b, 0]$, and corresponds to a line in $\mathbb{R}^3$ through $(0, 0, 0)$ and $(a, b, 0)$; in particular, it lies in the $xy$-plane. It follows that the line at infinity of $P^2$ corresponds to the lines in $\mathbb{R}^3$ through the origin which lie in the $xy$-plane.
what's difference between Projection plane and Projective plane?
I would say that "projection plane" describes its role. Namely, it implies that you are projecting from some higher-dimensional space (e.g. 3d) to that plane.
Conversely "projective plane" describes its structure. What kinds of points it contains, what axioms it satisfies. The fact that using homogeneous coordinates makes sense.
Your can do a projection from a 3d space onto a projection plane that also happens to be a projective plane. Doing so will allow you to preserve direction information for things "at infinity". You can even consider the map between 3d homogeneous coordinate vectors and their planar interpretation to be such a projection from a 3d space to the projective plane at $z=1$ using the origin as center of projection.
But the two concepts don't need to go together. You can do projection onto a plane while staying purely with an affine description of that plane. You can talk about the properties of the projective plane without any projection involved in the process. So from that perspective I'd consider the two concepts to be fairly independent.
Best Answer
Short answer: the two concepts “projective plane” and “projection plane” are different things, though they are loosely related.
Longer answer …
The “projective plane”, often denoted by $P^2$, is an abstract mathematical concept. It’s used in a field of mathematics called “projective geometry”. As the other answer explained, the basic idea is to represent each 2D point by a 3D line passing through the origin. The benefit is that this allows you to represent 2D points that are “at infinity”. You can use this technique with any plane.
The “projection plane” is a specific plane that’s used in 3D computer graphics. The points of a 3D object are projected onto the projection plane to produce a 2D image. Quite often, the projection plane has equation $z=1$ in some coordinate system.
People often use 4D (homogeneous) coordinates and $4\times 4$ matrices to represent the 3D-to-2D projection in computer graphics. This approach is not much related to the projective plane $P^2$, but it is somewhat related to projective 3-space, $P^3$.
Similarly, if you use 3D (homogeneous) coordinates to represent points in any plane, you are effectively working with the projective plane, $P^2$. But note that this is true of any plane. In particular, it’s true of the projection plane that you use in computer graphics, so this is the connection between “projection plane” and “projective plane”.
The main reason homogeneous coordinates are used in computer graphics is so that perspective projection can be represented by a matrix multiplication. But you don’t have to use matrices and homogeneous coordinates if you don’t want to —- the whole projection calculation can be done just using ordinary 3D coordinates. And this approach doesn’t involve $P^2$ or $P^3$ or any other concept from projective geometry.