Eventhough you have an (accepted) answer as provided by MattL. I would like to point out just another explanation of that LMS update equation.
Least Mean Squares adaptive filter algorithm is the most fundamental application of (statistically) optimum Wiener filter, where the algorithm indeed performs a steepest descend (gradient based) search of the optimum operation point; best filter coefficients that provide minimum (expected value of the) error square.
The simplicity of LMS comes from its replacement of theoretical gradient estimation with very simple practical ones. And the filter coefficient update equation (for real data) becomes what you have provided:
$$ \bar{w}(n+1) = \bar{w}(n) - \alpha ~ e(n) ~ \bar{x}(n) $$
Where $\bar{w}(n)$ is the $N$-tap FIR LMS filter coefficients (its impulse response) at time (or the iteration) $n$,
$$\bar{w}(n) = [w_0(n), w_1(n),...,w_{N-1}(n)]^T$$
and $\bar{x}(n)$ is the filter input of $N$ samples: $$\bar{x}(n) = [x(n), x(n-1),...,x(n-N+1)]^T$$ at time $n$ beginning with current sample $x(n)$.
The parameter $\alpha$ (actually $\mu$ in most textbooks) is the very important step-size parameters that controls the filter convergence behaviour vs resulting steady-state error power after convergence and the term $e(n)$ is the current (scalar) error at time $n$ given by the difference of current value of the desired response $d(n)$ and the current filter output $y(n) = \bar{x}^T \bar{w} = \bar{w}^T \bar{x}$ as $e(n) = d(n) - y(n)$.
$\bar{w}(n+1)$ is the updated (next) value, based on the current error computed via $\bar{w}(n)$ the current value of filter coefficients.
The equation you have provided just displays the vector equation above in terms of its individual elements.