From what I understand, 'både' is a conjunction and is often used with 'och.' So "I speak both English and Swedish" is "jag talar både engelska och svenska." Whereas 'båda' is a stand-alone pronoun ("Jag tycker om båda").
Here is a great article with explanation: http://www.thelocal.se/blogs/swedishteacher/2016/01/09/bade-or-bada/
Anyone, feel free to correct me if I'm wrong :)
You're right. både is always used together with och, it cannot be used on its own.
båda can be used either on its own or together with a noun, like båda böckerna 'both books'.
I thought people "har fel" and things "är fel". I quess I was wrong. So when is it "är fel" then? In this sentence can these sides mean e.g. sides of a cube or like different sides of conversation or both?
No, you're certainly absolutely correct, but in this case of sides you're really talking about people on either side of an argument, not about actual sides. We use har fel for opinions and statements, whether expressed by a person or e.g. a newspaper or a political party. If you were talking about e.g. the sides of a cube being the wrong size, it would indeed be är fel.
I wrote, "Both pages are wrong.", but it seems from the discussion that my answer should not have been accepted.
I'm actually fine with it being accepted - if the pages are wrong, you're typically talking about the opinions written on them, so it follows the same reasoning.
The ending looks like it would be The both sides...why am i wrong? Help please
Nope, doesn't work. That would be misstag in Swedish, and you gör or begår them, you don't har them.