Learn Icelandic: Hann biður um hjálpina.

Biðja is a strong verb with a slight irregularity. In the present tense singular (for I, you, and he/she/it), the j drops out and the ending becomes -ur. Therefore, "he asks" is hann biður.

Does the preposition um always translate to "for"?

Um most commonly translates to "about" or "around", but it depends on the verb it is paired with. The phrase biðja um is a fixed combination that always means "to ask for" or "to request".

Why does "the help" end in -ina here instead of -in?

The word for help, hjálp, is a feminine noun. If it were the subject of the sentence, "the help" would be hjálpin. However, the preposition um always forces the next word into the accusative case. For feminine nouns, the accusative definite article ("the") is -ina, making it hjálpina.