What is the difference between system memory and internal storage? I am using LG Optimus L3. 

When I look into the app manager, there is a system memory of 157MB and an internal storage of 1GB. I can't install all the apps on internal storage. It's something like internal SD card.

Can I somehow extend the system memory and shrink internal storage?
"System Storage" is where the OS and the System Applications are installed. This could also be considered where the User application are installed (based on your question). User installed applications are installed in /data/apps, settings and app data is stored in /data/data, and then application cache data is stored in /cache. These are all part of the "System Storage". 

Internal Storage is an area that is used like an sdcard. As far as the Android OS is concerned, it is an sdcard. Some applications may save data to it, but it is mainly used as direct storage for your stuff. Like pictures taken with the camera, music, ringtones, things like that. 

You cannot just adjust the size of system memory. Is it technically possible, yes. But doing so could require you to have knowledge of linux tool, rooting, custom roms, recoveries, and even possibly having to compile your own linux kernel and ROM or other information that I am not thinking of at the moment. Which is well beyond the scope of this site.