"My mom is angry."
Translation:אמא שלי כועסת.
It's an Aramaic word and in Aramaic the definite article is at the end of the noun instead of the beginning. It would be redundant to add the Hebrew definite article. Aramaic was the primary Jewish language for centuries from roughly the Persian period to the end of the Talmudic period. Hebrew has always been the sacred language, but there are good reasons that there is so much Aramaic in the Bavli. Anyway, Hebrew nouns that end with א are usually Aramaic.
You shouldn't add 'ה' to these words: אמא, אבא, סבא, סבתא.
It's not like "The my mom is angry", since that is - however badly it sounds in English - the way you usually say it in Hebrew. (for example, הבית שלי = my house, but literally "my the house" or "the my house").
The excuse I've heard is that they are from Aramaic, where they already have definiteness (so adding 'ה' would be double definiteness), but I'm not sure if it's true.
Words that end in aleph (see the examples radagast gives) are Aramaic and so you can omit the definite article as I explain above. However, some people still include the definite article, so it's a matter of learning to recognize that you may encounter both. For general rules for when to use the definite article, see a basic grammar such as Glinert.
It's actually the other way round - אמי is more formal. It's the possessive suffix, and can be used on any noun, but it is more formal than using "שלי". For example -
My house = הבית שלי/ביתי
My table = השולחן שלי/שולחני
My mother = אמא שלי/אמי
The reason you remove the 'א in "אמא", is because the word in Hebrew is actually אם. (the added 'א comes from Aramaic).