The reports hold a lot of useful information. You receive reports for trade, attacks, tribal invites and a few other things.
Reports are automatically sorted by category.
Reports for attacks and defences give you coloured dots. The follows explains what the colors mean.
The Fight did not cause you any losses. You won the fight but you suffered troop losses. You lost the fight. This was a scout attack.
Sometimes you won't need particular reports or you will get too many of them. In that case you can set up a filter. You can find that option under "Reports" "Filters". Once you have activated it, you will not receive any further reports until you uncheck it.
Reports are automatically deleted after a while. If you want to keep reports permanently you have to archive them. This option is only available for premium accounts.