You just need to ensure that your phone doesn't have the same application already installed whilst you are trying to install the same application's new APK.
So before installing the application, follow the following steps:
1. Go to Settings of your phone and look for an option that lists all applications in your phone and try to find whether the application that you are trying to install is already listed. If it is, then uninstall it from there.
2. Usually uninstalling by following step 1 removes the application from your phone however, if you are using a tablet, sometimes Android still keeps the APK file in the downloaded directory. So come out of settings and find out "All files" folder. Look for your app's APK in "downloaded" folder inside "All files" folder and remove it from there too.
Once you have deleted your app by following the above steps then you can go ahead and install your app. Hopefully you won't see "an existing package by the same name with a conflicting signature is already installed" error message.