OK.

Compression will produce smaller Illustrator files but will not affect output.

Illustrator files, by default, are saved as one file in two distinct formats, one the native Illustrator data and one a PDF. When you edit the file in Illustrator only the native Illustrator part of the file is used and the PDF portion is ignored. A New PDF will be created when the file is saved. Compression only affects this part of the file, but is lossless (unlike JPG compression). Compression usually has little effects on file sizes since it mainly acts on the vector data, which is usually smaller than the raster data and only acts on the native Illustrator data which is usually smaller than the PDF data. Small vector-only Illustrator files will see the most benefit, but are usually fairly small to begin with. Still, I usually ignore the Compression setting, which means I leave it on since that is the default.

Here is a simple vector file saved with compression on and off…

When you view the Illustrator file in Bridge, Windows Explorer, InDesign, or Photoshop you are accessing only the PDF portion of the file. This contains all placed and embedded raster art (like placed TIFFs JPGs, and PSD files) at full resolution, all vector art as vectors where possible (vectors have infinite resolution) and rasterized vectors and vector effects at the resolution specified under Effect > Document Raster Effect Settings.

Greater space saving can be achieved by making sure all placed images are linked, not embedded, that raster effects (shadows, blurs, and effect listed under Photoshop Effects) are minimized. If you do not need to view or output the Illustrator file in another program you can save with the Create PDF Compatible File option turned off…

I save seen files drop from a few hundred megabytes to about one megabyte. Really. You won’t be able to view those files in another program or place then in InDesign or Photoshop. But if you don’t need to anyway, then the space saving can be significant. The time saving is even more worthwhile, since saved will take a fraction of a second, not as much as a minute. This also means you will save more often since the wait for every save will be negligible.