1A standard polite expression used to apologize, or to get someone's attention (a stranger, a waiter, a clerk, etc.). Extremely common in daily life — covers everything from a light apology to a polite 'excuse me, can you help me?'
遅れてすみません、電車が止まっていました。
Sorry I'm late — the trains were stopped.
すみません、お水をもう一杯いただけますか？
Excuse me — could I have another glass of water?
あ、すみません！ 足を踏んでしまって。
Oh, I'm so sorry — I stepped on your foot.
すみませんが、駅までの道を教えていただけませんか？
Excuse me, could you tell me the way to the station?
2An expression of thanks tinged with humble apology — used when someone has gone to trouble for you, conveying both 'thank you' and 'I'm sorry to put you out.' A characteristically Japanese way to thank that sounds more polite and self-effacing than ありがとう.
わざわざ届けていただいて、すみません。
Thank you for going out of your way to deliver it.
こんなにいただいてしまって、すみません。
I really shouldn't accept all this — thank you so much.
席を譲ってくれてすみません、助かりました。
Thank you for giving up your seat — that really helped.
お忙しいところ、すみませんでした。
Thank you for your time even though you're busy.