GeoJSON is a format for geospatial data. It's based on the JSON format, and extremely popular in web mapping. Because it's a kind of JSON, you can parse GeoJSON easily in most programming languages.
GeoJSON is most similar, in terms of capabilities, to the Shapefile and KML formats: it can store both geometry and properties in the same file. Unlike KML, GeoJSON doesn't specify how to store styling information, so you can't store a "blue polygon", for example - you can only store "a polygon."
TopoJSON was inspired by GeoJSON and can be used to store some features more efficiently than GeoJSON itself.