Can't login to MacBook Air with username/password or with Apple ID credentials

I suddenly could no longer log into my MacBook Air after it went into screensaver mode. I tried my username and password, but it did not recognize. After many failed attempts (I also tried to restart it a number of times), it gives me an option to login with my Apple ID credentials. It also does not recognize this. Do you have a solution to this problem?

One question- Does the computer need to be on a network to use the AppleID credentials? Since I can't login, I can not get connected to the wireless...