I know right click and the right option there is save/load icon,but my question is,when I click the slot there is nothing happen,I can't save,can't input save name...

Might be a bug. I right click, click the save icon, then click a slot and just type, then hit enter when done. I just double checked, and there is *no* indicator for where the keyboard cursor is- rightn ow as I type this I have a | indicating where the next letter goes, some programs use a _ and Shadow of the Comet seems to lack either. Whatever save box has a yellow highlight is the one that my text shows up in while typing.

I do type in a name but then I get an in game error message "error while saving"

I get the same error. Has anyone ever solved this?

I don't know if this is still relevant to anybody - but since I just played this game for the first time and I had the issue with saving:
In full screen mode I didn't manage to save.
I then changed to windowed mode and it worked.

Savegames can be found here:
\Shadow of the Comet\cloud_saves\INFOGRAM\SHADOW.CD\

Great game!
It's pretty hard and lengthy at times though, in my opinion, so I recommend a walkthrough like the one from the-spoiler dot com.
Also YT has complete vids of the game.