Lattice Points in x-y plane

Question

1. What are Lattice Points?

2. Which points in x-y planes are Lattice Points?

   Is (m,n) a lattice point where m,n are any integers?

Answer 1

That is correct. The term "lattice points" usually refers to the points with integer coordinates.

Answer 2

No, that's not accurate. The points $(m,n)\in\Bbb Z^2$ are a lattice, but they are not the only lattice in $\Bbb R^2$, consider the sets:

$$\{(a,b\sqrt 2): a,b\in\Bbb Z\},\quad \left\{\left(a+{b\over 2}, b{\sqrt{3}\over 2}\right): a,b\in\Bbb Z\right\}\tag{$*$}$$

These are also a lattices.

Generally a lattice in $\Bbb R^2$ is a $\Bbb Z$ module of rank $2$ which contains a basis for $\Bbb R^2$.

As Cameron notes, this just means that you have integer combinations of two $\Bbb R$-linearly independent vectors from $\Bbb R^2$ (it's important that they be linearly independent over $\Bbb R$ and not something like $\Bbb Q$)