An 'eggshell' is the thin, white outer coating of an egg. The shell is very thin and breaks easily. So if you are 'walking on eggshells' you are in a situation where you could break something (or get in trouble or ruin everything) very easily.
Synonym: walking on thin ice
2) We're walking on eggshells with our landlord -- she told us that if we have one more loud party, she's going to kick us out of our apartment.