In the introduction to the chapter on Legendre functions, the DLMF starts off with the following notations
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_\nu(x)$, $\mathsf{Q}_\nu(x)$, $P_\nu(z)$, $Q_\nu(z)$; Ferrers functions $\mathsf{P}_\nu^\mu(x)$, $\mathsf{Q}_\nu^\mu(x)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_\nu^\mu(z)$, $Q_\nu^\mu(z)$, $\boldsymbol{Q}_\nu^\mu(z)$; ...
and, in particular, it draws a distinction between Ferrers functions and the associated Legendre functions. However, other resources tend to blur between the two, such as e.g. Wolfram MathWorld, which states
Ferrers' Function
An alternative name for an associated Legendre polynomial.
(where, of course, by "polynomial", the MathWorld page means "function which is sometimes a polynomial but normally isn't").
What distinction is the DLMF trying to make here, what are the resulting differences between the resulting functions, what does "on the cut" mean, and why is all of this necessary?
The functions of the first and second kind $P_\nu^\mu$ and $Q_\nu^\mu$ are typically defined via hypergeometric series. These functions can be analytically continued to the set $\Im(z)$ > 0 and to $\Im(z)$ < 0, but they have poles at the points $\pm 1$ which preclude the possibility of them being single-valued functions on the entire complex plane.
There are several obvious ways to obtain values of these functions for the real line. One is as follows:
$$P_\nu^\mu(x) = C_\nu^\mu \lim_{y\to 0^+} \left(P_\nu^\mu(x+iy) +P_\nu^\mu(x-iy) \right)$$
with $C_\nu^\mu$ an appropriately chosen constant. That is, taking in average of the boundary values of the analytic continuation to $Im(z) > 0$ and to $Im(z) < 0$. This gives the Ferrers' functions. Another possibility would be to define
$$P_\nu^\mu(x) = C_\nu^\mu \lim_{y\to 0^+} (P_\nu^\mu(x+iy)$$.
The notation distinguishes between which choice is made.