Genetic genealogy is the combination of traditional genealogy with the help of DNA test that focus on family relationships (eg. autosomal DNA tests from Ancestry, MyHeritage, 23andMe, Family Tree DNA and Living DNA but also Y-DNA and mtDNA tests).
There are two general approaches:
1) Using the output provided by your DNA testing company
23andMe eg provides the user with "Relatives-in-common" information which is showing who amongst your DNA matches has also inherited the same piece of ancestral DNA from a common ancestor. MyHeritage is also telling you who triangulates (see DNA Triangulation ) with you and thus confirms a common ancestor.
The other DNA testing companies are way more restrictive in the information they provide to their users on relationships. Ancestry.com is withholding most information but gives an estimate of a relationship to a DNA match.
2) Using raw DNA data provided by your DNA testing company
The DNA testing companies provide you with a link or process to download your raw DNA data that they have sequenced from your DNA test sample. Especially for companies like Ancestry.com this is the only way for end-users to identify where on their 23 chromosomes they triangulate with their DNA matches.