Which word order is correct?
Ich mag gern Katzen
Ich mag Katzen gern
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Both word orders are correct, but the meaning is slightly different.
Ich mag gern Katzen.
takes the fact that you're liking something as a given, a kind of "precondition". Then, this sentence adds the information what you're liking, namely cats.
Ich mag Katzen gern.
on the other hand has the "precondition" that you're talking about cats. But what about cats? The new information in this case is that you're liking them.
As an aside, the correct form of the adverb would be "gerne". "gern" is a casual shortening that's so common that you don't even mark the shortening anymore (so it's not written as "gern'" most of the time). But technically, it would need to be "gerne".