Example sentences
méi没guān关xì系，màn慢màn慢lái来。
It's fine, take your time.
duì对bù不qǐ起。méi没guān关xì系。
Sorry. It's okay.
About this word
没 is "not have" and 关系 is "relation, relevance", so 没关系 is literally "there is no relation". The sense is that what you are apologizing for has no bearing on anything between us. It is the standard answer to 对不起, in the way 不客气 answers 谢谢. Two usage points. 关系 alone means both "relationship" (我们的关系很好) and "relevance, consequence" (这没关系, "this does not matter"). And in casual speech 没事 is a close rival that can also wave off 谢谢 or a bump in a doorway, while 没关系 stays tied to apologies and to "never mind".
The story behind the character
没
没 is 水 (water) beside a shape that began as a hand reaching into water and got scribbled into something like 殳. Xu Shen defined it 没, 沈也, to sink, and 沈 is just the old form of 沉. Something that has sunk is gone, and gone is where méi, not have, comes from. Two readings, one idea.
关
关 is the traditional 關, which is 門 (a door) plus a bar. Xu Shen wrote 關, 以木橫持門戶也: a piece of wood laid across two door leaves to hold them shut. The door bolt. A bolt links two doors, so 关 grew into the link itself, and 关系 is a connection. Duan Yucai summed it up: 凡立乎此而交彼曰關, stand here and reach there, that is a 关.
系
系 is 糸, a thread of silk, with a hand shape on top. The earliest writing shows fingers holding a strand. Xu Shen read that top stroke as a sound element and defined it 系, 繫也, to tie. He got the shape wrong and the meaning right. 係 and 繫 both fold into 系 now, so one thread does all the work in 关系.