Learn Russian: Ребёнок спит на груди у мамы.

Грудь (chest) is a feminine noun that ends in a soft sign (-ь). These nouns belong to a special group called the 3rd declension. When used in the Prepositional case to show location (like after на), they take an -и ending instead of the standard -е.

Why use у мамы to mean "his mother's" instead of a possessive pronoun?

In English we say "on his mother's chest", but Russian frequently uses the preposition у followed by the Genitive case to show possession, especially when talking about body parts or things directly affecting a person. Literally, this translates to "on the chest by mom." You will see this structure often, for example in у меня болит голова (my head hurts).

The dictionary form of the verb is спать (to sleep). Why does it conjugate with an "и" as спит?

Спать is slightly irregular. Even though its infinitive ends in -ать, it conjugates using the typical endings for -ить verbs (the Second Conjugation), and it drops the -а- from its stem. That is why it becomes он спит (he sleeps) rather than "спает".

Could you also say Ребёнок спит у мамы на груди?