The Midway Point is a small bar of tape suspended between the Midway Point posts. When Mario breaks it, in addition to becoming Super Mario if he isn't already, he'll start from the Midway Point Entrance when he dies, instead of starting at the beginning of the level. Midway Points are most commonly found at the half-way point of a level, hence the name. If Mario clears the level and enters it again, he'll start at the beginning instead of the Midway Point.
In Lunar Magic's overworld editor, in Layer 1 mode, you can click on the "Midway point obtained" checkbox for a level. When you enter that level for the first time, Mario will start at the midway point. Then he'll start at the beginning for each subsequent entry. As useless as this might seem, Super Mario TKO uses this to great effect in the first world's castle. When you re-enter the castle, Mario will enter a destroyed, ravaged, and collapsed castle with a Toad near a door explaining that if you go in the door, you'll go back in time to before the castle was destroyed.
When using a midway point, don't use the bar itself directly from Map16 or it will re-spawn every time Mario enters the level, even if your using the midway point itself. Instead, use the midway point bar from the Extended Objects List. However, if you do this, part of the midway tile itself will disappear when using the midway point. So here's what you do:
1:Put in the Midway Point Bar from Extended Objects.
2:Overlay the part of the post that isn't the bar itself with Map 16 object 35.
Also note that if you put the Midway Point Bar from Extended Objects on a screen boundary, the tiles will become misplaced.