Data on Openfmri.org are typically stored in zip, tar.gz (also called .tgz), or tar.bz format.
To extract .zip data on Windows simply right click, and click 'extract'. To extract tar.gz/bz files, you'll need a program such as 7zip.
On linux/unix/Mac OS X you extract them from the command line using the following commands (all without the quotes):
For zip files: 'unzip filename'
For tar.gz files: 'tar -zxvf filename'
For tar.bz files: 'tar -jxvf filename'
For Mac OS X only, you can also often just right click on the file and click extract, however this may fail for certain filenames. The above commands will work if you have problems.
Please keep in mind that the dataset files are often several gigabytes, and thus they can take 5 to 10 minutes to extract.
All (uncompressed) datasetsare also available via Amazon Web Services S3 s3://openneuro/ This allows convenent downloading of whole datasets or arbitrary subsets (for example only T1s or only a particular task) using tools such as CyberDuck or AWS CLI.