Well, first off, すいません is the casual version of すみません. すみません/すいません actually has a lot of meanings. "Excuse me", "I'm sorry", even "thank you". It really depends on how it's used.
When you're asking for help:
すみませんが、図書館に行き方を教えてください。
"Excuse me, but can you tell me how to get to the library?
After stepping on someone's foot:
すみません。
"I'm sorry." ("Excuse me" works as well in a lot of instances of "I'm sorry")
When borrowing a pen from someone:
すみません。
"Thank you."
The thing is that in each case you're inconveniencing someone; taking their time, stepping on their foot, or borrowing something; and you're essentially recognizing your potentially negative impact on them.
As for the verb...well I think (please correct me if I'm wrong), that すみません is mostly idiomatic. Because according to jisho.org, 済む means:
1: to finish; to end; to be completed;
2: to merely result in something less severe than expected;
3: to feel at ease;
4: to feel unease or guilt for troubling someone; to be sorry
Which sort of has multiple, somewhat conflicting, definitions.