BRIK and HEAD files are medical imaging (generally, brain imaging) data files generated and used by the AFNI (Analysis of Functional NeuroImaging) analysis package. The files are multi-dimensional raster data, or "voxel" data -- in essence, pixels with a value in three dimensions. The AFNI format also supports time-series data, or multiple values at each voxel over time, per participant, or with different meaning (for example, a beta value and a t-statistic).
The .BRIK portion of the file contains the actual voxel data, while the .HEAD file contains the metadata needed to interpret the data.
The AFNI software suite is open-source, written in C, and available here.
Note that (nearly?) all AFNI programs support the much more widely-used .nii file format; converting data to that format may be a good move. A caveat is that the .nii header supports much less information than .head files.