From the Super Mario Wiki
Big Stars are Stars that appear in New Super Mario Bros.. They are only found in "Mario vs. Luigi" mode, and a set amount of Big Stars must be collected to win. Whenever a Big Star is collected, everything in the level is reset.
In the settings for the game, the host player will decide how many Big Stars are required to win the battle. Also, at the top of the upper screen, the amount of Big Stars a player has is displayed above his or her character.
One Big Star at a time appears in the level. After players have collected at least one Big Star, players can make each other drop Big Stars (if they have any) by jumping on them or attacking them with a Fireball, Shell Dash, etc. Players can drop Big Stars by touching an enemy, falling into a bottomless pit, or getting crushed by walls. Ground Pounding or Drill Spinning into a player can make that player drop a lot of Big Stars.