What is a semi-untethered jailbreak?
The semi-untethered Pangu jailbreak only works on iOS 9.2 to 9.3.3, and no, you cannot upgrade specifically to iOS 9.3.3.

But, to answer your question about what a semi-untethered jailbreak means, taken from the Jailbreak Subreddit,


  After rebooting your device, your tweaks will not function and your device will appear to not be jailbroken. To fix this, open the Pangu app, lock your phone and wait for it to [refresh].


A normal jailbreak as we know it is called an untethered jailbreak. Back in the day, there were also tethered jailbreaks. Tethered meant that every time you rebooted your iOS device, you were required to re-jailbreak with a computer (or else your device wouldn't even boot!). Untethered meant that the device can jailbreak itself automatically every time during the boot process. Semi-untethered is a mix between the two - it needs to be jailbroken every time you reboot, but it can be done by an app on the device instead of needing to do it with a computer.