# Above all, my brothers,

"This is important, my fellow believers" or "Especially, my fellow believers,"

# do not swear 

"do not make an oath" or "do not make a vow"

# either by heaven or by the earth

The words "heaven" and "earth" refer to the spiritual or human authorities that are in heaven and earth. 

# let your "Yes" mean "Yes" and your "No" mean "No,"

"If you say "Yes" or "No," say it truthfully"

# so you do not fall under judgment

"so God will not punish you"