The problem is, latest version of Skype requires Mac OS X 10.9 or later version. So you will not be able to use or any other latest apps like latest Safari, Chrome etc until you upgrade the OS X to 10.9 or later. The latest is 10.12 Sierra.
Follow the steps below to upgrade the OS X to latest macOS if your Mac support it.
Click on the Apple icon and App Store. This will bring App Store. Click Updates tab. If your Mac can handle latest OS X, then you will see macOS Sierra there with FREE UPGRADE button. Click Free Upgrade and enter your Apple ID and password.
This will download the update. It may take a few hours to download. Once the download is completed, it will bring the OS X Installer. Follow the onscreen instructions to install it. It may take up to 3 hours to install the upgrade.
If your Mac doesn't support latest 10.12 Sierra, then you can download and install 10.11 El Capitan. You can install El Capitan. You can download it from here --> https://itunes.apple.com/app/os-x-el-capitan/id1147835434?mt=12
This will take you to App Store with 10.11 El Capitan link. So click GET button and use your Apple ID and password ***** download it. Once it is downloaded, it will open the installer (will take a few hours to download) and follow the onscreen instructions to install it.
Once you upgraded to 10.11 or 10.12, then you can download latest Skype from www.skype.com
If you have any difficulty on following the steps above, please let me know and I will continue to help you.
Note : This is not my final answer. I strive to provide Excellent service or 5 star service. If you feel you've received anything less or if you are not satisfied or if my solutions didn't work, please don't rate me bad or poor or ok as it will badly affect my job here. Instead, please let me know and I will try my best to help you until you are fully satisfied or the issue is resolved.