how to make a updatable event calender ?

how to make a updatable event calender ?

Hey guys , i am just building a simple html website and basically the guy i am building this site for now wants a event calender to be added to the site . 

- Add new events and also 
- View existing events that are present. 

Something very similar or exactly like this  . 

I have no idea what to use here , i know jQuery UI may have a similar widget , but than can i add and view events. also i guess i will need somekind of a backend right ? 

Thank you.