@artyom
Raster tiles: A map image is 'cut up' into little squares: one set of image tiles per combination of style, language etc. This is replicated at all the required scales.

Vector tiles: The map data that would be required to display all the styles for that square is extracted from the database, simplified and stuck in an intermediate format. One set of tiles can support multiple styles, languages and angles that would be impractical with rasters.

@openstreetmap

They can be used to trimex a quantum position in space-time.

The article links to an explanatory page in the very first paragraph after the lede.

Vector tiles serve up maps as vectors: points, lines and polygons. They store geographic data (like what makes up OpenStreetMap) in a format that allows for dynamic styling and interactivity. For users, vector tiles will mean a new, modern-looking map style with seamless zoom on openstreetmap.org, the map can be updated more quickly when data changes, and it should perform better for users.