Right right right, as someone who speaks alright Russian, but more importantly as a native Serbian speaker (Serbian has the exact same "formal and informal speech" system as Russian), I'd really like to help.
So basically, "извините/извини" means both "I'm sorry" and "excuse me", we just use the same phrase in the situations where English speakers would choose between these two phrases. So that kinda makes it simpler, there's only one phrase for you to remember instead of two.
Now for the difference between "извините" and "извини": in Serbian, Russian and many other Slavic languages, there's a difference between speaking formally and informally.
In an informal setting, when talking to friends or family, you should use the second person singular to refer to them. So the equivalent of the English "you" would be "ти", "I'm sorry " would be "извини", "look, please" would be "посмотри, пожалуйста" and so on. The informal speech is also used by young people when talking to other folks their age, even if you don't know them, but usually NOT with people older than 30. It's not really a generation thing, it's just that you're taught to be formal with anyone you don't know well
In a formal setting, so at work, talking to people in the street and so on, you should use the second person plural to refer to them. So "you" would be "ты", "I'm sorry " would be "извините", "look, please" would be "посмотрите, пожалуйста". It's just the plural form of the same thing.
Hope this was helpful!