Question: I normally write using the *American* English forms, but for some particular words, I tend to naturally write it in the *British* manner (like with the word *favourite* rather than *favorite*).

Is it wrong to mix both styles? If not \*wrong\*, does it looks weird?

Answer: 
I think most people would recommend you stick to one style or the other. Why? Well, it doesn't matter too much, but if you mix styles the reader might notice! And that's bad—if they're noticing stuff like that, then they're paying attention to *how* you're writing rather than *what* you're writing. 

In other words, you're distracting the reader. You probably don't want that. You want them to focus on what you have to say, right? So unless you have a specific reason to mix styles, just stick to whichever style seems appropriate.
It must be noted, to add to the other answers, that some official exams will penalise you for mixing. For example, Cambridge (British English) will probably tolerate American English if you are consistent, but certainly penalise you if you mix them. I have also known of professors in the UK that consider American-spelled words an orthographic mistake, and therefore lower your grade for it (and being harsh!).