How to do a backup with time machine?

Hi Communitieee,
I dont know how to make a backup with time machine. Can anybody help me?
And i have another question.. whats wrong with library? when i start my mac the message is popping up but i dont know what it means..


Ok thanks mende1.
Whats with time machine? I already take a look to the link but i could not find the answer of my problem. Can you please take a look of this screenshot?!
I dont have an extern hard drive. i just want to save my pics and the other data of my mac.