If somebody buy something from my magento website, how can I know this? Is magento sending information e-mail about orders to admin? or we have to check from backend?
Magento has this built in as a default behaviour. To enable this navigate in the admin to System > Configuration > Sales > Sales Emails. In this section you will find options for all states of the order. In each section you can find the option "Send [State] Email Copy To". You can add your email address to this field to receive the notification.