3 answers
The "false-discovery rate" is the fraction of positives that are false positives at a given p-value threshold. It is a property of the threshold, not a property of a gene. So, if you do a 1000 tests, and get 100 positives at p<0.05, your FDR is 50% (as 50 false positives would be expected in 1000 tests at p<0.05).
Thus, technically speaking, it doesn't make sense to say a gene has an FDR. This is why many tools will use the term "adjusted p-value" or "q-value". The adjusted p-value is the FDR your experiment would have if you set the threshold at the p-value for this gene. Thus, in our example above, a gene with a p-value of 0.05 would have a padj/qvalue of 50% because if you set the threshold at 0.05, you would have a 50% FDR.
FDR is a statistical approach used to adjust your p-values for multiple testing. For example, in DESeq2 the default FDR method applied to your p-values is the Benjamini–Hochberg procedure.
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Benjamini and Hochberg (BH) defined the concept of FDR and created an algorithm to control the expected FDR below a specified level in a list of independent p.values. As mentioned by @andres.firrincieli DESeq2 adjust p.value by BH algorithm and @i.sudbery clearly explain and @ATpoint already answered the query well on point. padj is FDR.