Without the International Date Line, people who travel west around the planet would discover that when they returned home, it would seem as though an extra day had passed. This situation actually happened to Magellan's crew when they returned home after their circumnavigation of the earth.
Here's how the International Date Line works. Let's say you fly from the United States to Japan. Let's suppose you leave the United States on Tuesday morning. Since you're traveling west the time advances slowly thanks to time zones and the speed at which your airplane flies, but once you cross the International Date Line, it's suddenly Wednesday.
On the reverse trip home you fly from Japan to the United States. You leave Japan on Monday morning but as you cross the Pacific Ocean, the day gets later quickly as you cross time zones moving eastward in an airplace. However, once you cross the International Date Line, the day changes to Sunday.
The International Date Line is not a straight line, either. Since its beginning, it has zigzagged to avoid spitting apart countries into two days. It bends through the Bering Strait to avoid placing far northeastern Russia in a different day than the rest of the country. Unfortunately, tiny Kiribati was split. In 1995 the island country of Kiribati decided to move the International Date Line. Since the line is simply established by international agreement and there are not treaties or formal agreements associated with the line, most of the rest of the world followed Kiribati and moved the line on their maps. Most recent maps show the change and you'll see the big panhandle zigzag which keeps Kiribati all within the same day. Now eastern Kiribati and Hawaii, which are located in the same area of longitude, are a whole day apart.