Animating and Saving a Movie of a Function with “ListAnimate”

ClearAll["Global`*"];

Animated functions are a little different than graphics  of simulations. 
Here, we're just going to plot a sine wave. The frequency of the wave will change over some short time period. Since the frequency is what changes during the animation, you might think of the frequency as defining the frame number.

f[x_,ω_]=Sin [ω x];                                    (* the basic function being plotted *)
xmin=0;xmax=10;ymin=-1;ymax=1; (* these are plot limits *)
ωmin = 0.1; ωmax=2.0;ωstep=0.1 ;

So, there will be (2.0-0.1)/0.1 +1 = 20 frames in this animation...

picturestack=Table[Plot[f[x,ω],{x,xmin,xmax},PlotRange->{ymin,ymax}],{ω,ωmin,ωmax,ωstep}];

So here, "picturestack" is a "Table" ,which is Mathematica jargon for an array. In this case,
it happens to be an array of plots, and a plot is more or less the same thing as an image.
You should remove the semicolon in the previous line to see the array itself!

To pick out just a single frame of "picturestack", you give it a frame number, not a value of ω!
So, you might ask to see picturestack[[3]], for example. This would correspond to the plot having ω = 0.3.

setofframes=ListAnimate[picturestack] (* ...View the "movie" here in Mathematica *)

path = "C:\\Users\\pogo@geneseo.edu\\Desktop\\anim1.gif";

You'll obviously need to put in a path that's appropriate to your own computer!! 
Note: I don't know about Macs, which might need forward slashes, but this PC path & folder structure requires double-backslashes where normally only a single backslash is required.

The resulting file is an "animated gif". You can't double-click it to watch a gif, and you can't I can't view it in a "photo viewer" (it only shows the first frame). However,  you can drag it into PowerPoint, or into a web browser  to see the animated result.
Also note that the result shows borders etc. from Mathematica, which is probably not what we want!

Export[path,setofframes]; (* now save it to the previously named location! *)