I have a 13-inch, Mid 2010 MacBook and I am stuck with the version 10.11.6, I am not able to make any upgrade. Currently I am not able to download any app from the App Store (only old versions) and I am afraid, soon I will not be able to update some applications (as Unity, VS and GitHub Desktop). Could you help me with any solution? I do not really want to stick with old version on my applications. The replacing of my Mac is not an option
From the looks of the details on EveryMac.com your Mac will take up to macOS 10.13.x (High Sierra).
Apple's current macOS version is 10.14.x so you are actually in pretty good shape as far as that goes. High Sierra is still getting security updates and most released software will work on the last 2 to 3 macOS versions. Many will work on versions earlier than that.
You should be able to sign into the App Store and download macOS High Sierra. here is a link that will open that up in iTunes, https://itunes.apple.com/us/app/macos-high-sierra/id1246284741?mt=12
Once you have upgraded to High Sierra that will be the last major revision of macOS that is supported on your Mac. Depending on your needs and the apps that you use you may need to get yourself a newer Mac that supports macOS Mojave (10.14) and later. But IMHO you probably have a couple of years (at least) of life left in your current Mac.