Creature genes, both good and bad, are passed on from parent to child during breeding and occur randomly otherwise in player owned creatures. A creature can be born with multiple genes.
The only way to remove genes is by utilizing the gene lab in the Onyx City 2 area.
You can see the effects of various genes within a creature's status menu. Click its name in camp, then click the Genes button to see a creature's available genes.
Only player creatures are affected by genes, not enemies. Also, any creature generated before March 13th 2015 may not have any genes at all. I apologize if this inconveniences you.
Genes can be acquired in the following ways:
Your Gene Research Manual is located under "Player>Books>Genes". In order to complete this book, you must currently own a creature afflicted with the respective genes.
For example, if a Thunderbird you have possesses the Thundercrack Gene, the entry will be lit up and marked as Being Researched. If you later choose to destroy that Thunderbird, the Thundercrack Gene will be faded out again and marked as Sample Unavailable within your Gene Research Manual.
The benefit of researching a gene is that if it is positive, there is a higher chance it will occur in a newly born creature's gene pool. If it is negative, there is a lower chance it will appear. Genes passed on via breeding do not benefit from this bonus.
The maximum numbers of genes a creature can have varies by the method it receives them.