The Knowledge Hacker is used to gain knowledge from a world the AI controls. This makes the AI unhappy as a hacking response, but does not raise AI Progress. To use a normal science lab you need to remove the AI command station from the system first, and that increases AI Progress. So the purpose of the knowledge hacker is to gain more knowledge without increasing AI Progress. You can gain knowledge faster by bringing multiple hackers, but this will make the AI response nastier.
The Ship Design Hacker lets you pick which ship you get from the ARS. To use that, you need to put the hacker on the same planet with the ARS while the AI controls it, and leave it there for 10 minutes (a timer should appear on the hacker). At the end of that 10 minutes you can pick your design, but you still have to take the planet to actually get it. Doing this makes the AI angry. Note that if you bring any science lab to the planet you can see the available designs, which gives you a chance to see if you want to hack it or not (if you don't hack it you get the first one in the list). You can't speed this timer up by bringing more hackers (but having multiple ones there will make the AI response nastier).
There's one other type of hacking - About half the time, a map will have a building somewhere on it called the SuperTerminal. That is a link directly into the AI network. If you take control of the system it's in, you hack the SuperTerminal and reduce AI Progress by 1 every 15 seconds (while increasing the floor, or minimum value by 0.2). The AI will respond to this as well by spawning ships from the SuperTerminal, so it's a good idea to fortify the system with defenses and fleet support. The response starts off small and increases to crazy the longer you do it. To stop it you have to destroy the SuperTerminal with your ships.
All the types of hacking boost the AI's hacking response. If you use the Shpi Design Hacker, using another type of hacking later will start at a higher response level. If you sit on the SuperTerminal for an hour (reducing AIP by 240) you can expect a very violent reaction to trying to use the Ship Design or Knowledge Hackers. So scouting is key on these, and you have to decide what you're going to use it for. I recommend never doing knowledge hacking until later in a game, because early on you can take systems as needed and get knowledge that way instead. That leaves your options open later to choose where to place your hacking. (Ship design hacking may be worth doing early/mid game, if you scout an ARS and see a ship that you really want is an option. All ships are not created equal and it makes a real difference.) In particular, unless you have reasonably high AIP the SuperTerminal isn't worth using, as the Data Centers and CoProcessors will lower it significantly on their own.
(Now if you're playing something like Fallen Spire and taking 30+ systems, the SuperTerminal is awesome.)
