"I am sorry."
Sajnálom is another way to say I'm sorry. If you take it literally, the meaning is close to "I regret it" (referring to a bad thing that happened), and you can use it to express sympathy for something as well as apologizing for something you've done. Elnézést is strictly an apology - closer to "pardon me" or "excuse me" than it is to "I'm sorry." (If you want to break it down to a literal meaning, it's like "[I ask your] looking-away," although I don't think anybody really hears it like that.)
You can also feel sorry for somebody, saying "Sajnálom őt".
And "Elnézést" could be compared to someone "looking past" some mistake. "I am willing to look past this mistake" - "Hajlandó vagyok elnézni ezt a hibát". But in general usage "Elnézést" is just "Excuse me".
I have never heard this statement in conversation. Bocsanot, bocsi, elnezest...are all common