"Tú miras las fotos de mi cumpleaños."
Translation:You are looking at the pictures of my birthday.
Why not? My birthay's photos
What is the difference between My birthay's photos and birthay photos?
Which is the rule?
This is a very good question. "The photos of my birthday" and "The photos from my birthday" would be much more common.
You could probably say "my birthday photos" and it would be okay. Sometimes we use nouns as adjective in English such as birthday cake or birthday party. Birthday photos is probably not as common but is understandable.
Take a look at this wordreference link and you can find many other examples of birthday as an adjective.
On the other hand, my birthday's photos doesn't sound correct to me. Perhaps it makes it seem as though the birthday is the owner of the photos? I'll have to think about whether there is a rule about this. In the meantime, here is a link about possessives. It says, "The possessive form is used with nouns referring to people, groups of people, countries, and animals." Of course, there are probably exceptions but this will get you started.
Puedo explicarlo en español también si quieras. However, I've noticed in other threads that your comprehension of English is quite good.
I am so tired of seeing this same statement! I have memorized it, but i don't think that is helpful towards learning the language.