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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. A commonly-used version used for display is recognizable for having "sawtooth" borders (projecting triangles) toward the north and south, the whole projection bordered by straight edges. In addition to display, versions of the projection are also used as the basis for file formats for storing data relating to the surface of a sphere (e.g., from intensity mapping).
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.