When you look for something, you search for it trying to locate or find it. When you look at something you know where it is and you see it.
As you know, ‘look at’ and ‘look for’ are phrasal verbs which there are a lot of in the English language while in most other languages there are no phrasal verbs at all. In German ‘to look for’ means ‘suchen’ and in Russian ‘искать’.
Now, you are ready to answer this question: look for vs. look at