对不起 (Duì bù qǐ) means I am sorry.
对 (Duì) means right or correct. You may remember the Chinese word by considering X with a horizontal line on top along with a + (like) sign and a dot with it in the middle.
不 (bù) means not or negative. You may remember this by considering three fibers hanging down from the roof.
起 (qǐ) means to rise or to start. You may remember this word by considering the L shaped character with two horizontal lines in the middle with a 2 shaped character in the right.
Literally , 对不起 means Right not to rise.
没关系 (Méi guānxi) means That’s all right.
没 (Méi) is an adverb meaning not. You may remember this by considering left side with three dots, one downward dot and two upward dots, and the right side with X shaped character with a horizontal line above in the down and in the upper position a car like character.
关 (guan1) means to close or to concern. You may remember this by considering a small insect with a pair of three legs.
系 (xi4) means system or relation. You may remember this by considering nine different lines; six lines moving from right upper to left lower side and two lines moving from right lower to left upper side and a vertical line.
关系 (guānxi) is a noun meaning relation.
Literally, 没关系 means not relation.