The quickest way is to click on the dropdown to the right of your calendar name under My Calendars and choose Share this Calendar.
If you want the calendar public you would select Share this calendar with others and then one of the two options under that (Make this calendar public or Share this calendar with everyone in the organization). If you do not want it public, make sure those checkboxes are empty.
To share with a specific person choose what Permission Setting you want. There are four (see descriptions below image):
Make Changes AND manage sharing - this should just be the owner of the calendar, as anyone with it can delete the calendar.
Make changes to events - allows editing of the calendar
See all event details - read only version of the calendar
See only free/busy (hide details) - can only see times taken, cannot see what is happening during those times or access anything regarding them.
Then enter the persons email address in and press enter, repeat for each person you would like to share it with (don't forget to change the rights if some should only have See all event details rather than Make Changes to Events or vice-versa.
Click on Save when you have added everyone you need to. Should you ever need to remove or change rights for someone, just go back to Share this Calendar, change the type of access to the right of their name and save.
To remove them, click on the trash can to the right of their permission settings and click on Save.