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.

If the last row (the all zeros row) is removed, then the matrix you get is the transpose of the parity check matrix of the binary Hamming code.  
