I'm a few weeks into learning Korean and I learned how to write statemnts and questions via (습니 까 & 습니다). We take verbs, drop the -다 and apply the correct formula. However, my textbook also showed a bunch of verbs that end in 요, for example: ' 아/여/어 요'. How do these work? Is it the same or something more advanced? In summary, how would you say 'Are you laughing?'?
It's about formality but really depends on the relationship: customer/clients, boss or subordinate, higher in rank, respectful or disrespectful, rude or polite/formal.
~습니까?, ~습니다 is very formal. When you go to a bank and talk to a banker, they may speak this way to you. It gives the feeling of keeping some distance and strictly for a business relationship, and treating you as a client/customer. And also to someone higher in rank.
~요?, ~요 more widely used, not used between close friends but closer than business relationship - it's gentler and friendlier.
~어?, ~어 use with close friends, close and same age or close AND younger, lovers, or sometimes boss to subordinates in work place. It is considered to be very rude if one talks this way to a stranger, business relationship, someone older, higher in rank. You may hear this in TV/movie, police officers speak this way to criminals (talking down, disrespectful), teachers to students in the class (lower in rank).