Those 1-click rooting apps will function on fewer and fewer devices, and that's a good thing, because they work by exploiting security holes that shouldn't exist.
The traditional way to root is to install the root manager (e.g. SuperSU) via a custom recovery. That requires a custom recovery, of course, which in turn requires an unlocked bootloader (most modern Android devices will have a locked bootloader nowadays).
This is how you unlock the bootloader:
- Open the Android Settings app, scroll to "About the tablet/phone" at the very bottom, tap it.
- Find "Build Number" and tap it 7 times. That should unlock the Developer options.
- Tap the back button to go back to the main screen of the Settings app.
- Developer options should be near the bottom now, tap it.
- In the Developer Options screen, activate "Enable OEM unlock" (if present) and also USB debugging (USB debugging enables ADB, the Android Debug Bridge, which can do some useful stuff).
- Reboot the device to bootloader / fastboot mode.
This can be done via ADB, by connecting the device to PC via USB, and then running the command "adb reboot bootloader" (adb is a command line utility which needs to be present on your PC. I won't go into detail how to install or use it, there are enough guides on the web).
- After a few seconds, the device should reboot and launch the fastboot mode that is part of the bootloader. You can test if it worked by running the command "fastboot getvar all" (device still needs to be connected to PC via USB, and "fastboot" is another command line utility you need to have on PC). If the command outputs a bunch of text with details about the device, then it's working. Among the text should also be some info about the device's lock state, e.g. "(bootloader) unlocked: no".
- Finally, run the command "fastboot oem unlock" and follow the instructions on the device screen to unlock the bootloader. Be aware that this will erase your data partition for security reasons, i.e. all of your apps and media files will be wiped!
Note: some manufacturers might slightly change the unlock command, e.g. add a device-specific security key that you first need to obtain from them ("fastboot oem unlock secret_key_here").
Once the bootloader is unlocked (or if it wasn't locked to begin with), you can launch a custom recovery via "fastboot boot img_file", with "img_file" being the filename of the custom recovery tailored to your device.
However, not all devices support the "fastboot boot" command. In those cases, you need to flash (install) the custom recovery via "fastboot flash recovery img_file", and then boot it before you reboot Android - because when Android starts, it'll normally automatically replace the foreign custom recovery with the original stock recovery.
The command "fastboot oem reboot-recovery" works on some devices (MediaTek), but not others. If it's not supported, there's probably some key combo you can hold to make the device boot the recovery instead of Android (e.g. the keys power+volume_up, or power+volume_down).
Once in the custom recovery, you can simply install the latest SuperSU zip, or the root manager of your choice, and your device should be rooted.
The main hurdle in all of this is getting a suitable custom recovery for your device. Someone has to build it first, but that can be tricky. If you're lucky, you might be able to use an existing custom recovery of a device that is very similar to yours (same chipset), but chances for that aren't good.