A plane is like an infinitely big sheet of paper that continues forever in all directions. It's sort of like a single slice of the three-dimensional world in the same way that a line is like a single slice of a sheet of paper.
If you want to get philosophical about it, we say that the universe we live in is three-dimensional, each slice of that (each plane) is two-dimensional, each slice of those slices (each line) is one-dimensional, and sometimes just to be ridiculous we say points are zero-dimensional.
Planes go on forever like lines do. We can't draw planes that extend forever and ever, so we use borders to represent them. Unlike lines, we name planes with super-fancy cursive capital letters. You may want to brush up on your calligraphy skills.
Sometimes it's hard to tell from a picture whether a plane contains a given line or a point, so we try to make the drawings very obvious.
Here, plane 
contains line l and point A, but not line m or point B.
Any three points can be contained in a single plane. To visualize this, imagine a triangle connecting the three points, and simply extend the boundary of the triangle to get a plane. However, if you have more than three points, they might not live on a single plane. A square pyramid is an excellent example of this.
The only plane containing four points is the bottom of the pyramid, and that plane will never contain the vertex of the pyramid. When a bunch of points lie on the same line, we call them collinear, so it only makes sense that when a bunch of points or lines lie on the same plane, we call them coplanar.
Sample Problem
Suppose we have a bunch of collinear points. Are they coplanar?
It might seem like we don't have enough information to answer this question, but it turns out that we do. No matter how many points we had, we know for a fact that we can capture collinear points by a single line.
That means if we choose any plane containing that line (and we have infinitely many to choose from just by rotating the plane around the line), that plane must contain all the points. That means the collinear points are also coplanar.
Mathematicians sometimes abbreviate this phenomenon by saying something like "collinearity implies coplanarity" or "collinearity is stronger than coplanarity." To be fair, it has been working out a lot recently.