That adapter is a splitter, it doesn't turn a jack into a headphone and microphone. Headphone jacks send signals out, to the headphones, while microphone jacks accept incoming signals, from the microphone. There are combination jacks, which your computer might have, but you'd need to either look up the details on your specific laptop, or give them to us here so we can check that.
If your laptop does not have a dedicated microphone jack, you should be able to purchase a USB microphone to use instead.
UPDATE:
From the product website (http://us.acer.com/ac/en/US/content/model/NX.MGRAA.002) we learn your device has a built-in microphone, probably right near the built-in webcam.
There is not information specifying which sort of headphone jack it is. However, there will likely be icons next to the headphone jack indicating whether it supports just headphones, or if it's a combo jack that supports microphones as well.
If it is a combo connector, you should be able to purchase a combination headphone/microphone such as are sometimes used for cell phones. If it is not a combo unit, and you do not want to use the built-in microphone, you'll need to purchase a USB microphone as I suggested earlier.