Learn Finnish: Sairaan lapsen äiti on täällä. - The sick child's mother is here.

Adjectives in Finnish have to match the case of the noun they describe. Since lapsen (child's) is in the genitive case to show possession, the adjective describing the child, sairas, must also be in the genitive case, becoming sairaan.

How do we get from the dictionary word sairas to sairaan?

Words ending in -as or -äs undergo a stem change before you add endings. You drop the -s, double the vowel, and then add the ending. So, sairas becomes the stem sairaa-, and adding the genitive -n gives us sairaan.

Why is it lapsen and not lapsin?

Many older Finnish words ending in -i change that -i to an -e- when you add grammatical endings. The basic word is lapsi (child), the stem becomes lapse-, and adding the genitive -n makes it lapsen.

Could I say äiti sairaan lapsen to mean the mother of the sick child?

No, in Finnish, the owner in the genitive case almost always comes right before the thing they own. You must put the owner first, saying sairaan lapsen äiti (literally: the sick child's mother).