Typically, local regression (loess / lowess) is used to create smooth plots.
Assuming the points are equidistant along the X axis, what's the advantage of using local regression compared to a simple moving average with an appropriate window size (which is obviously much faster)?
A simple smoothing average can be interpreted as a local linear regression with a rectangular kernel. A rectangular kernel assigns equal weights (read importance) to each point falling within its kernel support (read window). If you think this assumption encapsulates your modelling assumptions adequately then you have no reason not to pick a simple moving average for smoothing. If you think this assumption is a bit oversimplifying... read along.
Let's assume that we look at data $(y_i,t_i)$ but actually what is going on is that $y_i = y_{\text{true}}(t_i) + \epsilon_i$ where $y_{\text{true}}$ has a some odd but smooth parametric form and $\epsilon \sim N(0,\sigma^2_{\epsilon})$. By smoothing we try to estimate $y_{\text{true}}$.
We could go ahead and fit a model across all data; something like: $y = \beta_0 + \beta_1 t + \epsilon$ (or a higher degree polynomial) but we suspect that this is too restrictive. We have the implicit understanding that data close to a time-point $t$ are more relevant to the value $y_{\text{true}}(t)$ than data further way from $t$. So we decide to built a window around $t$, say $[t-b, t+b]$ where $b$ is a bandwidth. Now, if the assumption is that all points within $[t-b, t+b]$ are equally important to estimate $y_{\text{true}}(t)$ then a rectangular kernel where all point are weighted the same is perfect for us. But maybe we think "... within the window some central points matter more" and we try another kernel) (eg. triangular or Epanechnikov) that assigns higher importance to central points. Or actually we are not really certain about the assumption of a window to begin with so we fit try a kernel (eg. Gaussian) that has infinite support. ($b$ is always to be estimated using cross-validation). Local linear regression gives the ability to test and actually incorporate all these assumptions to our final estimates for $y_{\text{true}}$.
Finally let me point out that "lowess/loess" are utilising locally weighted linear regression to smooth data but their are just one type of the local polynomial methods (eg. the Nadaraya–Watson estimator, one of the earliest estimators of this kind) used in semi-parametric regression. Other models (eg. roughness penalty methods, like spline smoothing) are also available; see A.C. Davison Statistical Models, Chapt. 10.7 for a nice concise introduction.
Moving average is what you get when you are Using a zero degree polynomial [which] turns LOESS into a weighted moving average. Higher degrees yield different answers.
