There are a number of ways to say sorry in Russian, all depending on what sort of emotion you need to convey.
Suppose you have done something wrong and admit your guilt — then ‘prosti’ and ‘izvini’ said to a close friend or family member or ‘prostite’ and ‘izvinite’ if talking to a person you are less familiar with are the most natural and appropriate ways to say sorry in Russian.
But if feelings of guilt are overwhelming, the best choices for someone eager to say sorry in Russian are ‘prosti, pozhaluysta’ or ‘prostite, pozhaluysta’ and ‘proshu proshcheniya’.
The younger generation is also extensively using the word ‘sorry’ itself, especially on the Internet, but it is rather colloquial and should be avoided in more formal conversation.
Life is life, and situations in which you may need to say sorry in Russian or any other language are unavoidable, and even though you are now equipped with the most common ways to say it in Russian, it is always best not to do things you can later feel guilty about.