You can only have a maximum of four of ANY one card type, with two exceptions. For example, you can have four Potion Trainer cards maximum.
For Pokémon, you can have only four of a certain species. For example, you can only have a maximum of 4 Ampharos cards — this includes EX and Lv. X. I don't know about Break because I stopped playing before they came out, but I assume they work the same way. If someone could fact check this for me that'd be great.
Finally, the two exceptions: you can have as many Basic Energy cards as you want. Note that this does not apply to Special Energy cards, like Double Colorless Energy or Rainbow Energy. You can only have four of those maximum.
The second exception is Arceus cards. You are allowed to have as many Arceus in your deck as you please.
DISCLAIMER: That's how the rule was a few years ago. I'm not sure if the unlimited Arceus rule still stands, but I don't see why not.