Select the chart and edit the data source. The order of the series in the left hand side of the dialog will determine which series show in front of others, as long as they are all plotted on the same axis. Select a series and click the up or down arrows to move its position.
In the screenshot you can see the Series order for the lower chart. Series1 is listed after Series2 and will be plotted on top
Edit: Apparently the option to re-order the series does not exist in the respective dialog on Excel for Mac. In this case, edit the formula for the data series in the formula bar.
Select the data series in the chart and look at the formula bar. You will see something like
The last number in this formula is the order of the series. You can edit and change it in the formula bar. The lowest number series will be plotted behind higher number series.
Before: the orange series has the number 2 and is plotted on top of the blue series (which has the number 1)
After: the orange series has the number 1 is plotted behind the blue series (which now has the number 2)
So, to bring a series to the front, make sure that its order number is the highest of the series in the chart.