HEALPix (which stands for Hierarchical Equal Area Isolatitude Pixelisation) is a type of map projection that can be used for a map of Earth (i.e., one alternative to the Mercator projection) or any spherical surface, such as the celestial sphere. It is recognizable for having "sawtooth" borders toward the north and south, i.e., projecting triangles. It is used for display and also is used as the basis for formats for storing the data over the 2D of a sphere surface, and also is the name of one of the pieces of software for producing and using the format and projection.
HEALPix is not the projection commonly used to display the cosmic microwave background (or other intensity maps) as an oval: that is a Mollweide projection.
HEALPix is also the name of software that the deals with the HEALPix projection and also data formats (HEALPix format) used by the software and for some astronomical projects.