"Staggered" refers to the jagged edges of isometric maps that have an overall rectangular shape. These maps emphasize the north/south and west/east axes, and often have North up (example: Civilization 2). Diamond maps on the other hand emphasize the diagonal orientation and movement. North is often at the top right (example: Simcity 2000). Also notice the different coordinate systems:
"Normal" or "diamond" isometric map:
Notice the overall shape of the map is a diamond.
"Staggered" isometric map:
Notice the overal shape of the map is a (rough) rectangle
See also this explanation:
The math in terms of rendering both types of maps is similar but has its differences. Diamond maps are a bit more tricky particularly in terms of optimizing the render loop.
Diamond maps are typically used in RTS and Tactical Combat types games whereas Staggered/Block maps tend to be used for RPG's and Turn-Based Strategy games.
In terms of the Tiled XML, staggered adds a new possible value for the attribute "orientation" of the map.
Finally, don't confuse these staggered maps with staggered square maps. The latter are a way to get pseudo-hexes:
Note how the tiles are squares, but movement and adjacencies conform to those of a hexgrid.