It all started a few months ago when someone committed a crime in Taiwan and fled to Hong Kong. To prevent HK from becoming a safe haven for criminals, the Chief Executive of HK proposed a new law to facilitate extradition of these crime suspects from HK to various jurisdictions in the region, including Taiwan and mainland China.
The proposed law even explicitly stated that it's not applicably to crimes political in nature. But some HK people were nevertheless concerned that it might be abused by China to target political dissidents in HK.
So they have taken to the streets to protest that law. As a result, the law was quickly suspended before it had a chance to pass, and a few weeks ago the HK Chief Executive officially announced the withdrawal of the law.
However, despite the concession from the HK government, the protesters pressed on, demanding four more concessions from the government, chief among them universal suffrage, or the direct election of the HK Chief Executive, who up to this point have been nominated from a narrow pool of Beijing-approved candidates, then voted on by a committee.
It's not entirely clear that China even had anything to do with the proposal of the law which started this ordeal. But the protesters have been shrewd to paint a picture, to great effect, of big bad China stomping on the poor helpless people of HK.
What I cannot stress enough, is the rampant violence and destruction from these protesters, which has done this great city, and many innocent citizens, unimaginable harm. Feel free to support their peaceful protests, but please don't simply pile on and encourage these violence and destruction.
(EDIT: If anything I said is untrue, please correct me. Use the truth to argue your side, don't be a coward and hide behind your downvote.)