Learn Italian: Ho comprato un regalo per il mio fratellino. - I bought a gift for my little brother.

It is the noun fratello (brother) combined with the diminutive suffix -ino. To stick them together, you drop the final -o of fratello and add -ino.

I learned that singular family members do not take the article with possessives. Why does this sentence say il mio fratellino instead of just mio fratellino?

Great catch! You usually drop the article for singular family members, like mio fratello. But there is a big exception: when you alter the family word by adding a suffix like -ino, the rule breaks and the article comes back. That is why it must be il mio fratellino.

Does fratellino only describe age or physical size?

Not necessarily! Diminutive suffixes like -ino and -etto are frequently used just to show affection. You could use fratellino as a term of endearment even if your brother is a fully grown adult.

Just to review, why is it ho comprato (I have bought) and not sono comprato (I am bought)?

Comprare (to buy) is an action that takes a direct object, because you are buying something (in this case, un regalo). Verbs that act on a direct object almost always use avere as their helping verb in the past tense.