Explain Like I'm Five

A white hat hacker (also called penetration testers) has permission from the target to hack them. And if they succeed in hacking them, they write a report about how they did it, this allows the target to fix the security vulnerability, and therefore avoid getting hacked in the future. The white hat hacker of course gets paid for this service. Because they have permission, this is completely legal. Some companies also offer "bug bounties", this is essentially the company offering an open invitation for anyone to hack them, as long as they promise not to break or steal anything, and to also report their findings to the company.

A black hat hacker does not have permission from the target, they break in and steal data, break things, or sometimes just gather information. They can then black mail the target, sell the data or try to monetize their services in many other ways. This is of course illegal.

Fiinally there are grey hat hackers. These people blend the above two. These are people with noble intentions, but they don't have permission from the targets, but they still report their findings anyway, or in some cases even fix the security issues themselves. This is still illegal, because they don't have permission, but in many cases the target doesn't want to press charges, because the grey hat hacker tried to help without expecting to get paid. There was a case many years ago where some regular printers has a massive security vulnerability, which allowed hackers to break into the network the printer was on, and use the printer to hack other things. And since no one ever patches their printers, they were left vulnerable. A grey hat hacker took it upon themselves to break into all the printers and force the printers to update, fixing the vulnerability. And since the people who had the printers didn't have any idea about the vulnerability or the hack, they didn't press charges... Needless to say, grey hat hacking is very dangerous, because your good intentions, and benevolent actions might still land you in jail if the company or person you are trying to help doesn't like it.