Is this a correct way to think of a subspace? A subspace in $R^n$ is just a set with conditions that create a hyperplane (so a line, a plane, etc.) in $R^n$?
And of course it has to pass the subspace test. 
But if what I said is correct, because a subspace has to contain the zero vector, so that the scalar multiplication test passes, does the line or plane HAVE to cross the origin? What's the difference between a line passing through the origin and one that is just shifted one unit up so it doesn't touch to origin? What makes the first a subspace and the second not?
If any of my questions were hard to follow I think it's because I have a poor understanding of what a subspace is so any help on that would be nice. Thanks!
 A: To define a subspace let us first define a vector space. A vector space (over $\mathbb R$) is a set $V$, such that for any elements $v,w\in V$ and $a,b\in\mathbb R$, we have that $av+bw\in V$. I.e. I can multiply a vector by a number and get another vector in my space, and I can add arbitrary vectors together and get a vector in my space. (See https://en.wikipedia.org/wiki/Vector_space for the full list of definitions).
Now, a subspace of a vector space is a subset of $V$, $W$, such that $W$ is a vector space. So, as you said a line through the origin is a subspace, but a line that does not pass through the origin is not a vector space. This is because it does not contain 0. Since we can multiply vectors by arbitrary numbers in $\mathbb R$, we can multiply any vector by zero. So every vector space must contain 0.
Now, if we have a finite dimensional vector space $V=\mathbb R^n$ and a subspace $W$ of dimension $m$, then one can show (using https://en.wikipedia.org/wiki/Gram–Schmidt_process) that there will be an orthonormal basis $\mathbf e_1,\dots,\mathbf e_n$ of $V$ such that $\mathbf e_1,\dots,\mathbf e_m$ is a basis for $W$. Then the $n\times n$ matrix
$$
P=(\mathbf e_1,\dots,\mathbf e_m,\mathbf 0,\dots,\mathbf 0),
$$
gives $m$ linear equations that define $W$. Namely 
$$
W=\{x\in V|Px=x\}.
$$
