Does this matrix have a name? (Maybe in combinatorics?) Or rather, is there a name for the class of matrices that resemble this one:
\begin{bmatrix}
  1 & 1 & 1 \\
  1 & 1 & 0 \\
  1 & 0 & 1\\
  1 & 0 & 0\\
  0 & 1 & 1\\
  0 & 1 & 0\\
  0 & 0 & 1\\
  0 & 0 & 0 
\end{bmatrix}
I made the matrix because I wanted to be able to think of each column as representing a thing that could be either chosen or set aside (like in combinatorics?), and each row as representing one way of choosing/not choosing from among the three things.

I don't think there's a standard name for this matrix, but the idea that it represents is well known and useful. Good for you if you invented it yourself.
Each row of your matrix describes one of the $2^n$ subsets of an $n$ element set (you have $8$ rows since $n=3$) using a string of $n$ bits, each either $1$ or $0$, indicating which of the $n$ elements of the set (these are the columns) are or are not in the subset.
The rows are sometimes called "bit vectors". If you think of each bit vector as a binary number then you've nicely labeled all the subsets using the numbers from $0$ to $2^{n-1}$. In your example you listed them in descending order from $7$ (the whole set) to $0$ (the empty set).
You can then do boolean algebra (union, intersection, complement, symmetric difference) with arithmetic on bit strings.
Some set counting problems are best approached with this representation.
