The complete rules for correct usage of articles are so many and detailed, that it's outside of the scope of this website to explain them all. It would take a large book.
But for the three examples you give, I can explain the rules.
- "It is a rainy day."
Generally, "a" means "one of many". There are many rainy days, and this is one of them.
- Sunil is the best student in our class at present.
"At present" is an idiomatic expression that does not follow normal rules for articles. You have to learn this expression on its own.
- This is my umbrella.
Words like "my/your/her..." are possessive adjectives. You cannot have both a possessive adjective and an article on the same noun.