Could be "Péter sees a nice house." for another possible solution as well. "beautiful" means more like "gyönyörű" in hungarian, while "nice" could also means "szép". These words are synonyms.
Why can`t it be: Peter sees one pretty house? I know that egy is a/an, but here it sounded also nice with one...
It's accepted now, thanks for pointing it out! However, the translation you're suggesting is a bit unnatural for the Hungarian sentence. The reason is that when you say something like Peter sees one pretty house. your highlighting the number in a sense, more than you do if you just use a house. This kind of highlighting would lead to a different word order in Hungarian:
Péter egy szép házat lát.
where the object comes right before the verb, and when pronounced, egy gets some additional stress.
Very hard to hear that there is an "egy" in this sentence. Does is sound different when it is between words?
As an indefinite article egy is not stressed, that's why it can't be heard so clearly. Keep practicing and soon it'll be easy to recognize it.
Also, Hungarian has a lot of assimilation. Before a word starting with a voiceless consonant, egy will become pronounced as ety and the plosive sound of the "gy" (devoiced to "ty") will not usually be released ... it will be a closure of the tongue and then the next consonant simply begins. I think this makes Hungarian sound beautiful, but it does make some sounds surprisingly subtle and you'll have to tune your ear for them.
Like in English, lát ‘see‘ and néz ‘watch’, ‘look at’, do not quite mean the same thing. lát is more general, it means simply to perceive something using your eyes. néz refers to a more active process of observing something.