Introduction
HEALPix is an acronym for Hierarchical Equal Area isoLatitude Pixelation of a sphere. As suggested in the name, this pixelation produces a subdivision of a spherical surface in which each pixel covers the same surface area as every other pixel.
Purpose
The original motivation for devising HEALPix was one of necessity.
Satellite missions to measure the cosmic microwave background (CMB)
anisotropy  NASA's
Wilkinson Microwave
Anisotropy Probe (WMAP),
and currently
operating ESA's mission
Planck
 have been producing multifrequency data
sets sufficient for the construction of fullsky maps of the
microwave sky at an angular resolution of a few arcminutes. The
principal requirements in the development of HEALPix were to
create a mathematical structure which supports a suitable
discretization of functions on a sphere at sufficiently high
resolution, and to facilitate fast and accurate statistical and
astrophysical analysis of massive fullsky data sets.
HEALPix satisfies these requirements because it possesses the
following three essential properties:
 The sphere is hierarchically tessellated into curvilinear quadrilaterals. The lowest resolution partition is comprised of 12 base pixels. Resolution of the tessellation increases by division of each pixel into four new ones. The figure below illustrates (clockwise from upperleft to bottomleft) the resolution increase by three steps from the base level (i.e., the sphere is partitioned, respectively, into 12, 48, 192, and 768 pixels).
 Areas of all pixels at a given resolution are identical.
 Pixels are distributed on lines of constant latitude. This property is essential for all harmonic analysis applications involving spherical harmonics. Due to the isolatitude distribution of sampling points the speed of computation of integrals over individual spherical harmonics scales as ~N^{1/2} with the total number of pixels, as opposed to the ~N scaling for the nonisolatitude sampling distributions (examples of which are the Quadrilateralized Spherical Cube used for the NASA's COBE data, and any distribution based on the symmetries of the icosahedron).
Applications of HEALPix to data processing and visualization have now spread well outside the original CMB field, as can be seen from the Resources and Gallery pages.
Main Features of HEALPix Software
The HEALPix software is available in C, C++, Fortran90, IDL, Java and Python.
In each of these languages are available an extensive library of HEALPix specific tools
and their supporting routines as well as a suite of programs implementing the
key features of HEALPix. Each package allows all of, or most of, the following (and more):

Spherical Harmonics Transforms:
 Fast simulation and analysis of fullsky maps of CMB temperature and polarization anisotropy (sky maps preview) up to subarcminute angular resolution
 Filtering of sky maps with arbitrary circular window
 Constrained and nonGaussian realization facilities
 Highly optimised Spherical Harmonics Transforms library (libsharp) used by all implementations for better performance
 Forward and backward scalar and spinweighted Spherical Harmonics Transforms
 Programs to manage, modify and rotate spherical harmonic coefficients of arbitrary maps

Pixel manipulation:
 Pixelation of the sphere supported down to a pixel size of 0.4 mas (milliarcseconds), corresponding to potentially 3.5 10^{18} pixels on the sphere
 Pixel queries in discs, triangles, polygons and strips
 Programs to search the maps for pixel neighbours and extrema of a random field
 Median filtering of sky maps
 Mask processing facilities
 Support for multi resolution maps (aka, Multi Order Coverage maps or MOC)

General:
 Comprehensive documentation (PDF and HTML); webbased and email support
 Automated installation and build scripts
 Most critical routines are parallelized

Visualization:
 Visualization facilities available in C++, Fortran 90, IDL, Java and Python
 Facilities to output HEALPix maps into Google Earth/Google Sky compliant images and into DomeMaster format used in planetariums.

Input/Output:
 routines to manipulate and visualize the FITS files generally used for I/O
For a detailed list of the new features added in the latest HEALPix release, see
the Getting HEALPix page.
For third party implementations in other languages (including Matlab/Octave and Yorick), see this HEALPix wiki page.
For third party extensions of HEALPix capabilities, see the Resources page.