"I am sorry!"
"Sajnálom" is more formal, and used is specific situations only. It is more like expressing sympathy than just saying a quick "sorry". You would not say "sajnálom" when you just accidentally bump into someone on the street. But if you bump into them and they fall over, you would say something like "ne haragudjon, nagyon sajnálom" while you are helping them to get up.
No. Literally, "bocsánat" means "forgiveness". So the strictly literal meaning of "bocsánatot kérek" (which is also commonly used), is "I am asking for forgiveness". Apart from that, "Vagyok bocsánat" is also in the wrong word order. The correct word order would be "bocsánat vagyok". That would mean "I am forgiveness" and it would still be wrong. Sorry.... :-)
Hi cosmo, "bocsánatOT vagyok" is grammatically incorrect, so it would not be accepted anywhere. "Bocsánat vagyok" (= I am forgiveness) may be accepted in poetry, the same way "I am the wind" or "I am hope" would be accepted. :-) "Ne haragudjon" literally means "don't be angry/cross" , in formal language. Informal would be "ne haragudj". You might find that your Hungarian friends will open a sentence with "ne haragudj, de... " before they tell you something that will make you upset. (something like "don't be cross with me for saying this but ..." ) :-)