Exclamation marks are always an indication for quests. The marks can appear above a person's head, above a black board at the stables- but also in the fields (these ones are secret quests).
All you need to know about the different colors can be read at "Symbols".
You have to do the quests otherwise you can't move on in the storyline and can't progress in the game. Exploring new places, meeting new people, getting money, items and reputation points is only possible by doing the quests.
To accept a quest you have to approach the person or item with the exclamation mark and click it. Normally a dialog appears which you have to follow. You then should accept the quest by clicking "Yes". You have the opinion to cancel the quest, but that won't get you further and the quest will be still available then.
If you don't finish an accepted (active) quest, you will have to start over after your next login.
Notice boards give you the following quests:
- Mucking out the stables
- Feeding horses
- Watering horses
By the time you can explore the whole of Jorvik there will be many daily quests to do.