The Black Panther Party began as an organization to police black communities and protect them from police officers. This involved following cops in black communities at a distance of ten feet, and when police officers interacted with (usually meaning trying to arrest) black folk, the Panthers would spend the whole time advising the black person on their legal rights and the cops on their legal limits. Keep in mind that they would do this openly carrying shotguns, which was perfectly legal at the time. That being said, of course it was going to be unpopular with a lot of people, legal or not.

When California saw the ways in which the Panthers were challenging the authority of the police, they moved to ban open carry in their state. This led to the famous protest by the Panthers in Sacramento right outside of the capitol, armed with an array of firearms and dressed in their famous attire. Needless to say, the Mulford Act (which took away the right of citizens to open-carry) was passed by the state Congress and then ratified by governor Reagan.

Of course, it would be a lie to say that the Panthers never did anything more than just follow police officers around. There were altercations, some deadly, and many were instigated by members of the Panthers. Bobby Seale was one of the Chicago Eight arrested for the Chicago Riots in 1968 before his case was separated from the others, and Huey Newton was charged in 1967 with having killed a police officer, though Newton's case saw several retrials and he was eventually freed. There was a famous case in 1969 where three members of the Black Panthers chapter in New Haven tortured and killed a member who they believed was an informant. Over time they resorted to things such as drug dealing and extortion which began to erode their support in the black community (quite ironic since one of their earliest programs was the establishment of drug and alcohol rehabilitation centers in black communities).

It's important to understand though that the Black Panther Party was not just another black power group. They were Maoists and fully dedicated Communists. They supported arming the black community to protect them from a police which they saw as a colonial force. Whether the Panthers were following the law to the last letter or not they were bound to be on the bad side of the government, and the FBI took out all stops to try and silence the Panthers. There are records of COINTELPRO plotting to disparage members of the Panthers, and quite famously in 1969 the head of the Black Panther chapter in Chicago, Fred Hampton had his house raided by agents after being drugged with barbiturates by an informants. The police went in guns blazing and shot repeatedly through the walls. He was shot multiple times, including twice in the back when he was on the floor. They also worked to disrupt their programs focusing on providing poor black children with breakfast since they saw it as propaganda for children. They would harass workers for the program, and in a raid on the Chicago office set fire to the third floor of their building, where they held all of the food for the kids.

Sources:

Black Against Empire by Joshua Bloom

Revolutionary Suicide by Huey Newton (autobiography)

The Black Panthers Speak by Philip S. Foner

The Murder of Fred Hampton (Documentary)

Edit: Fixed typo and gave links to books I referenced.