There is not much special about DNA data that would warrant a specialized compression algorithm. The most space-efficient representation of nucleotides represented as a binary file encoded in 2-bits, or a bit more versatile in .2bit format keeping headers and also 'N' characters.
The Burrows-Wheeler-Transform has been used in some alignment tools lately, and is
used e.g. in bzip2. Afaik, bzip2 is one of the best (in sense of compression) general-purpose compression algorithm around. Like any other lossless compression it can be also used for nucleotide data in FASTA and .2bit format.
For obvious reasons there is pretty limited use for lossy compression.
In case you want to compress multiple genomes of one organism, you should have a look at Genome Differential Compressor and it's paper.
Using GDC 'a whole human genome can be stored in less than 3.12 MB.' Of course, one has to store reference as well.
The answer naturally depends on your definition of "best" and on what data you need to compress. Typically specialized sequence compressors provide the best compactness, but general purpose compressors are often faster. (However, both of these tendencies have exceptions).
How much compression you achieve will depend not only the on the archive type you create, but on the application you use to compress it and the settings you use. If you want to compress something to use as little space as possible, you should definitely use 7z. You can even crank up the compression settings to save even more space, although it will take longer to compress and decompress. Most compression tools have settings to allow you to achieve a higher compression rate at a compromise of slower compression/decompression times and more RAM usage.
GTZ is a more recent compression technique that published significantly better performance for fastq compression than other methods - pigz, LFQC, Fqzcomp, LW-FQZip, QUIP, & DSRC2. Wondering how it compares to the other methods mentioned here like 7z and DELIMINATE.
Has anyone used GTZ and can speak to its performance?