What is a simple/intuitive way to explain a projection.
You mean "orthogonal projection"? If we suppose "a" is a nonzero vector in $\displaystyle R^n$ and "x" happens to be any vector in $\displaystyle R^n$, then the orthogonal projection of x onto span{a} is written as $\displaystyle proj_a x$ and is defined using this formula: $\displaystyle proj_a x = \frac{x.a}{\| a \|^2}a$
The projection vector can be called "vector component of x along a".
Edit: If you consider the operator $\displaystyle T: R^2 \rightarrow R^2$ that "projects" any vector $\displaystyle x \in R^2$ onto a line through the origin, by dropping a perpenicular to that line, it's called an orthogonal projection. This is the definition of projections onto lines through the origin of $\displaystyle R^2$.
Since you say "intuitive", imagine a light shining on a vector from far away. The shadow of the vector on a plane or line would be its "projection" on the plane or line. That is where the term "projection" is from.
Also, when you separate a three-dimensional vector into x, y, and z components, those are its projections on to the x, y, and z unit vectors, respectively. Since you can always choose a basis containing any one (non-zero) vector, you can think of the projection of vector u on vector v as one "component" of u in a basis containing v.