Geometrically, what is the span of vectors? Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?

Assuming it makes sense that the span of a single vector is a line, we can imagine the two vectors in 3-space.  Because the span of each vector lies within the space of each of them, we can draw the two lines that are in the direction of these two vectors:


*

*if the two lines are equal, then this is all of the span.

*otherwise, we can imagine picking one of the lines and sliding it along the other line such that it stays parallel to its original placement.  
To make the above intuition precise, the point is that the span of a single vector is the result of performing scalar multiplication on that single vector: that is, contracting it, shrinking it, or even flipping its orientation (making the direction go the other way).  Then we can get any other point of the plane by finding the projection of the point onto the first line, and then translating by the projection of the point onto the second line; this is why when the lines are the same we do not leave the line.
As far as the formal definition of the span goes, the span of a set $S=\{v_1,\ldots, v_n\}$ of vectors is given by the set
$$\mathrm{span}(S)=\left\{ \sum_{i=1}^n{c_iv_i} \mid c_i\in \mathbb{F}, v_i\in S\right\}$$ where $\mathbb{F}$ is the field that you're working over (likely the real numbers $\mathbb{R}$).
In the case where $S=\{v_1,v_2\}$, we're looking at the set of vectors of the form $c_1v_1+c_2v_2$.  Now, if $v_1$ and $v_2$ are in the same direction, then there is $c$ such that $v_1=cv_2$, so we can rewrite this as $c_1v_1+c_2v_2=c_1v_1+c_2cv_1=(c_1+c_2c)v_1$, which is just any element in the span of $\{v_1\}$.  This is why it is possible for the span to be a line.
Now, when they aren't in the same direction, they lie precisely in a unique plane.  Any point in the line that goes along the direction of the vector $v_1$ can be written as $c_1v_1$, and any point in the line that goes along the direction of the vector $v_2$ can be written as $c_2v_2$.  Every point in the plane will then be of the form $c_1v_1+c_2v_2$ by projecting the point onto $v_1$ and $v_2$ to find $c_1$ and $c_2$, respectively.  
