So, in the past week or two, during games, I'll randomly get a pop up saying attempting to reconnect. (In game, mind you.) It's like I've disconnected, or something, when I haven't.
It isn't my internet, I know that.
I heard something about a 'drophack'.
Whatever it is, I want to know what's going on. I'm losing matches, and getting blamed for it because of this.

I don't know what the "drophack" is, but I've been having this problem since the patch last week. It is even happening in custom games with nothing but bots. My ping is normally 72, and in multiplayer games it will go all the way up to about 14000. In custom games with all bots it will still hit around 1300, which is enough to cause a disconnect. I found closing the game client, relogging in and reconnecting fixes it sometimes, but with the tribunal sitting on their Cray supercomputers throwing bans around like they were candy to anyone who hasn't spent 2 grand on their system I wouldn't recommend leaving a game. Finish what you can, uninstall, and wait until Riot stops blaming the customers and starts looking into the issue.