Thoroughly means to do something so completely and with so much detail that nothing or very little could possibly be remaining. For example if you lost your keys and in an attempt to find them back you check every inch of the house, looking under beds, couches and going through all the drawers and cupboards. Once there is nowhere left to check then you have searched thoroughly. Or another example if you like computer games would be to complete the game, unlock every achievement, complete every side quest, find every piece of loot, etc. Once the game has nothing left you can say that you played it thoroughly. Thoroughly does not need to mean that is has been done to absolute completion, for example when searching for keys you don't need to pull up the floor or look in the roof or in the example of the game there could still be a few unlooted items. As long as a very good attempt at completion has been made then you can say it was thorough.
Another common use would be to say that you enjoyed something thoroughly, which would mean that you enjoyed it completely or would be like saying you extremely enjoyed it.
Throughout has a very different meaning and means that something is happening intermittently during a period. For example if I was to say "It's common for people to say your name a few times throughout a conversation" then the period would be any given conversation and the even which is happening multiple times during this period is the mentioning of your name. A simpler way of explaining this is that throughout means it happens at multiple places within the context of discussion. For example I could say "There is mess all throughout my house" and you would know that my house is very untidy with mess in multiple places.
I hope this helps.