If the number of points in an affine plane is finite, then if one line of the plane contains $n$ points then: all lines contain $n$ points, every point is contained in $n + 1$ lines, there are $n^2$ points in all, and there are a total of $n^2 + n$ lines.
First my confusion is how do we get that each line has n points. In class, my teacher asked us how many points the line has and someone said infinite but that was incorrect(why?). I got this definition from Wikipedia, but my teacher gave us that the affine plane has $n^2$ points and deduced that each line has $n$ points, but I can't follow that. I also need guide on how there are more lines than points. They say the euclidean plane is an affine plane and in this plane we say that lines have infinite points, to my understanding.
Beware of the diagram
First off, be aware that the terms “point” and “line” are just words used in the axioms. For some affine planes, namely $\mathbb R^2$, they coincide with your intuitive concept of what a point resp. a line is, but don't let that intuition lead you to assumptions about other cases. In particular, even if you draw diagrams using real lines, then you still can not view the collection of points in $\mathbb R^2$ which constitute these lines as points on these lines in your finite plane. Only those points which actually are points in your finite plane lie on these lines. This is likely the misunderstanding where the answer about the infinite number of points came from. So remember: diagrams are just a tool to visualize some combinatorics, don't treat them too literally.
See also my answer to your other question which seems based on the same misunderstanding.
Counting
You can argue about the counting like this. The proof is not yet fine-grained enough to use a single axiom in each small step, but you should be able to fill in the gaps. You'll mostly have to add the appropriate uniqueness arguments.
The number of points on a line is the same for all lines. Call that number (or cardinality) $n$.
Proof: We'll compare the number of points on two arbitrary lines $a$ and $b$. Choose $A_1$ on line $a$ and $B_1$ on $b$ arbitrarily but different from the possible point of intersection between $a$ and $b$. Then consider the line $c_1$ joining these two points, and all the lines $c_i$ which are parallel to $c_1$. Each such parallel will intersect both $a$ and $b$ in exactly one point each, i.e. $A_i$ and $B_i$. (For exactly one $i$, these two will coincide, unless $a\Vert b$.) This gives a bijection between the points on $a$ and those on $b$, so both lines must have the same number of points on them. This is the standard way used to compare cardinalities, so it works for the infinite case as well.
There are $n+1$ lines through every point.
Proof: Let $P$ be your point, and $a$ be a line not incident with $P$. Then every line through $P$ will either intersect $a$ in some point $A_i$ or be parallel to $a$. There are exactly $n$ points $A_i$, and there is exactly one parallel, so you have $n+1$ lines through $P$.
There are $n^2$ points in total.
Proof: Choose any point $P$. There are $n+1$ lines through $P$. On each of these lines, there are $n$ points: $P$ itself and $n-1$ others. Each point which is not $P$ will lie on exactly one such line. So the total number of points is $(n+1)(n-1)+1=n^2$: the number of lines times the number of points on each excepting $P$, plus $P$ itself.
There are $n^2+n$ lines in total.
Proof: There are $n^2$ points. Through each of these, there are $n+1$ lines, which amounts to a total of $n^2(n+1)$ incidences (i.e. point-line pairs). Each line has $n$ points on it, so it contributes $n$ of these incidences. Therefore you can get the number of lines as $\frac{n^2(n+1)}n=n(n+1)$.
Infinite case
For infinite affine planes, you are talking about infinite ordinals. $n=\lvert\mathbb R\rvert=\mathfrak c$ for $\mathbb R^2$ to be precise. Now $\mathfrak c^2+\mathfrak c=\mathfrak c$, so in this case, there will be the same number of lines and points, i.e. the same kind of infinity. See also this post for more details on $\lvert\mathbb R^2\rvert=\mathfrak c$.