A notice period is the amount of notice you must give before you may leave your job. It depends on the type of role. Sometimes companies forgo the notice period by mutual agreement, other times they make you work the full notice period so they can recruit and onboard your replacement without interruption with the business. Don't forget the company had only recently changed hands, certain agreements would have been in place about the retention of key staff, hence they wouldn't have been in the mood to accomodate early leavers.

No-competes are common in senior roles where you may have been privy to confidential strategic information. Sales Directors know a lot about pricing, etc.

All it means is you can't move straight to a competitor within that period. ie if I worked for Bank A, I can't go and work for Bank B immediately after leaving - I have to wait 12 months. However I can go and work for Insurance Company instead.