Imagine putting a 2d rectangle into 3d space with some orientation. Starting from some corner of the rectangle, you have two sides coming out from it. In 3d space, those sides make 3d vectors. You can generalize it to 3 vectors making a cube with magnitude (volume). And these don't have to be perfect rectangles and cubes, they can be parallelograms and parallelopipeds (3d parallelograms) and higher dimensional analogues.
Think of a scalar. It has a magnitude but it doesn't have a direction. It's 0-dimensional.
Think of a vector. It has a magnitude (the size of the vector) and it also has a direction, which points in a straight line through the origin. It's 1-dimensional. For example, the vector (2,0,0) has magnitude 2 and points along the X-axis. You could write that as 2 * x, if x is the vector (1,0,0).
A bivector also has a magnitude, but instead of being 0-dimensional (like a scalar) or 1-dimensional (like a vector, it's 2-dimensional. So you could have a bivector that "points" along the entire XY-plane (remember: two-dimensional) and has some magnitude, say, 5. You could write that as 5 * x * y, if x is (1,0,0) and y is (0,1,0).
If you attach physical units to these things, then you might have units of meters for vectors, and square meters for bivectors.
Having an understanding of subspaces in linear algebra is helpful.
"The area of the plane will depend on the length of the pencils". Surely the area of the plane is infinite? The area of the _parallelogram_ will depend on the length of the pencils.
And I can't see how "you can assign an orientation to the plane" other than by changing the directions of the pencils. Again this description sounds like it refers to the parallelogram, not the plane.
And I don't know what a rotor is.
But other than that, I'm doing great.
A vector is an oriented (+,-) magnitude(length) _in_ a line. A Bivector is an oriented (+,-) magnitude(area) _in_ a plane.
That area does not have any particular shape.
