In the card above you can see a portrait in the bottom-right-corner. This means that someone has been added to the card because it requires the person's attention.
When you add someone to a card it will show up in the assignee's To Do list. If they complete the card in the To Do, this will be reflected on the card as well in form of a green tick.
If you want to add someone to a card just open the card and press the Members button. At the top you will se a Suggested field. This field contains the last user who you added to a card, the most used user in that column and most used user on that board. If the user you wish to add is not present in the Suggestions field, just start typing the name of that user and it will search the entire user base of your organization.