The lockboxes come in the colors Green, Blue, Red, Silver, Gold and Sinister, where green Lockboxes are quite common and gold and sinister lockboxes are very rare.
To open a locked box you will need a Lockbox Key or Sinister Key for sinister boxes, which are sold in Shop. These keys can be purchased using Coins that are found free within the game, or using June Cash.
The contents of the boxes vary greatly between types, but it is also random, which means when the reward of lower level boxes can be good as well as the upper level boxes. A locked box can contain more than one item, which is usually between 1 and 3.
Check the page of each box for possible rewards in each: