I’ve seen the "big-$M$ method" referred to in different ways. What is the "big-$M$ method" and why does it seem to mean two different things?
People do use the term "big-$M$ method" to mean two different things. In both cases, the name refers to the use of a large constant, often denoted $M$.
The first use of the term refers to a method for finding an initial feasible solution for the simplex method. (Another common method for doing this is the two-phase method.)
Sometimes people also use the term "big-$M$ method" to refer to the use of big-$M$ parameters in logical constraints in (mixed-)integer programming problems. For example: if $x$ is a continuous variable and $y$ is a binary variable, then the constraint $$x \le My$$ says that if $y=0$, then $x$ must equal 0, and if $y=1$, then the constraint places no restrictions on $x$. Note that in this case, it’s important not to choose a value of $M$ that is too large.
Aside: I highly recommend @prubin's excellent blog post on big-$M$ in all its forms.