It depends: The meaning is the same for both in general conversation assuming the person still wants to be a teacher but the correct one is using "has". Over time, with informal English and with contractions it has become: "She´s always wanted to become a teacher" and finally "She always wanted..." if you use it without "has", you are talking in the past tense so its likely that the person no longer wants to be a teacher. eg> "Angela was such a nice woman, it´s a pity she died so young, she always wanted to be a teacher (now she won't be)" or "When I was young I always wanted to be a teacher (now I´m a butcher)

I am not native English speaker but I think I can answer your question here. The first sentence is correct the second is not. She has always wanted to be a teacher ---> She wanted to be a teacher for a long time ago and she still want to be a teacher. The time expand from the past until the current time. The second sentence is wrong because it means that she only wanted to be a teacher for a short specific period of time in the past. this conflicts with the always word. I hope you find this helpful. and I hope my explanation is right :)

Really it should be the first one. Eg: She has always wanted to be a teacher. (present tense) She had always wanted to be a teacher. (past tense) It is informal and not grammatically correct to say 'she always wanted to be a teacher.' As Joseph said, it is a contraction. People drop the 'has' 'had' in conversation. 'She has always wanted to be a teacher' --> 'She's always want to be a teacher' --> 'She always wanted to be a teacher'. So in written work, you would never drop the 'has'. It is better to keep the 'has' in speech as well, but if you drop it it doesn't matter. I hope this helps.

They are both correct, but the meaning is slightly different. In both cases, 'always' means something like 'from when she was very young', but the time frame is different. "She has always wanted to be a teacher", and the contracted form 'She's always wanted to be a teacher" are examples of the present perfect, and they tell us something about the PRESENT situation. 'She has always wanted..' describes a continuing situation. She still wants to be a teacher now. "She always wanted to be a teacher" is past simple, and it describes a PAST situation. Perhaps, as Joseph suggested, it might mean that she died young, before she could achieve her ambition. Or it could mean that she now does a different job - perhaps she wanted to be a teacher, was forced to go into another profession for some reason, and then gave up her dream of being a teacher. Or it could even mean that she actually did become a teacher. If you interpret 'She wanted to be a teacher' as 'She dreamt of being a teacher', we use this phrase to describe the situation before she realised her ambitions. The state of 'wanting' - in the sense of hope and ambition - finishes when you actually achieve your dreams. For example, 'She always wanted to be a teacher, and the day that she graduated from training college was the happiest day of her life.'