Publishing to a website/publishing target.
By default, using the default TBBs, the publisher will download the images and store them on the content delivery server. The image will not be stored in the content manager or publisher and the image will not be re-sized.
This is Tridion, so of course you can change this behavior by not using the default TBBs or implement a TBB which changes the default benhaviour. I.e. you could leave the external URL to the CDN intact instead of downloading the image and publishing it to the presentation server.
Preview in CME
Same behavior as in publishing, but the image will not be stored on the contend delivery end but in the content manager sever in order to be able to run the preview.
The content manager explorer will show a thumbnail of the image. This thumbnail is stored in a cache on the CME. The image is for CME use only, it is not accessible though an API and it is not available on the content delivery. The thumbnails will be stored in [Tridion installation folder]\web\WebUI\WebRoot\ThumbnailCache.
Other Tridion modules
As an example, the connector for SDL Media Manager has a TBB which is executed before the Default Finishing Actions TBB. The Media Manager Connector TBB will resolve the multimedia component links (i.e. in an image tag) for specific multimedia component schemas. This way the links are already resolved before the Default Finishing Actions will attempt to download the image from the CDN.
SDL Media Manager is SDL's Media Asset Management system which uses a CDN.