It is a just a list containing about 750 hard and interesting problems from past OIs of different countries. You can use this to track the problems you solve.

White Cell: Means this problem is available, but you haven't attempted yet.
Red Cell: Means that this problem is not available to submit. You may find the statement in the given link.
Black Cell: Means that this problem is not available, or in case of POI, translation doesn't exist.
Yellow Cell: Means that you've read the problem statement and currently trying to solve the problem.
Blue Cell: Means that you already know the solution to the problem but don't intend to implement it. However the count of these types are added to solved problems for the status bars.
Green Cell: Means that you've solved the problem successfully.

Go to the "My Checklist" page. Then, by clicking on a cell (outside the problem links), you can toggle the state of a problem. After you are finished click the save button on the top of the page.

Go to "My Checklist" page. Around the top of the page, you can find a static link to share your checklist.

They are some statistics about the problem, collected from official ranklist. "Perfect Scorers" is the number of contestants that got perfect score in that problem on main contest. And the "Points Scored" is the percentage of score that was scored on that task, in other words, (total score on that task) / (total score on all tasks) * 100%. Most of the time this statistics is taken from all participants. However, for some old contests, only ranking of medalists is available, in such case the statistics is only from those.