Repair occurring at start up

I am a complete novice to Wintel systems. I purchased a new Dell with (2)
half-terabyte hard drives which supposedly mirror all updates to my system.

A majority of the time, when I have closed all applications and performed a
full shutdown, on the next startup, I get a message that I did not shut down
properly the last time. Then an icon appears on the extreme lower right with
two small disk drives, one updating the other. This lasts for some time and
then I get a message to the effect that my system is OK. I can still work
while all of this is going on.

Does anyone have any idea what is happening here? Does it involve my
mirrored hard drives? Is there something I should be doing prior to shutdown?

Any help would be appreciated. Thanks.

Russ