Merge To Center merges any selected vertices so they become shared and positions the resulting shared vertex at the center of the original selection. Any faces and edges associated with the originally selected vertices also become merged as a result.
The Merge to Center feature merges to a central point based on the center of the current selection. If you want to collapse edges on a component by component basis, select the edges, then select Edit Mesh > Collapse, or shift + right-click and choose Merge/Collapse Edges > Collapse Edge from the marking menu.