Learn Russian: У меня есть хорошая новость.

In English, 'news' is an uncountable noun. In Russian, новость is a standard countable noun meaning 'a piece of news'. Because the speaker has one specific piece of good news here, it is in the singular form. The plural form is новости.

How do we know the gender of новость?

Words ending in a soft sign (-ь) can be either masculine or feminine, so you usually have to memorize their gender. However, in this sentence, the adjective хорошая gives it away! The -ая ending tells us that новость is a feminine noun.

Can you break down how у меня есть means 'I have'?

Russian expresses possession differently than English. Literally, this translates to 'By me there is'. У is a preposition meaning 'by' or 'at', меня is the pronoun 'me', and есть means 'is' or 'exists'.

Is it possible to drop the word есть in this sentence?

Not in this case. While Russian sometimes drops есть when describing the attributes of something you have (like 'I have blue eyes'), you must keep есть when stating the simple fact that you possess an object or an abstract thing like a piece of news.