I sometimes see and myself use sentences like. "Reminds me that..." rather than "It reminds me that...". Is it correct use? If so, in which type of sentences can we omit subject (like it)?
This is called conversational deletion, subject pronoun drop, or informally, "diary drop" because diaries are often written in this style.
It is usually used only in informal conversation and writing. As the name "subject pronoun drop" suggests, you can delete a pronoun when it is the subject and if it is clear from context. For example:
What did you do today?
(I) went to the beach.
What did you give him?
I gave (him) the book.
This is technically incorrect grammar, as the sentence has no subject.
But in informal speech we use technically incorrect grammar all the time. As long as the listener understands the intended meaning, there's no problem.
Al: "Did you give Bob something yesterday?"
Charlie: "The travel form."
Charlie's answer is not a complete sentence. But in context, it's pretty obvious that what he means is, "Yes, yesterday I gave Bob the travel form." So the extra words are not really needed.
Reminds me ... When I was in high school, most of the teachers insisted that answers we gave on tests must be complete sentences. So like if a guestion was, whatever, "What is the capital of France?", if a student just wrote "Paris", the teacher would mark it wrong. You had to write, "The capital of France is Paris." My chemistry teacher apparently didn't know that other teachers were demanding this and one day expressed frustration at the wordy answers we gave on his tests. He plaintively said to the class, "If I ask, what reaction occurs when you mix [chemical A] and [chemical B]?" (I forget what the example was now), "Just tell me what chemical is produced. You don't have to write, 'yes indeed, when you mix these two chemicals there is a reaction.'"