This post is basic biochemistry that we probably all know. I've just stuck it down as I've slogged through it while I've been working through Chris Masterjohn's treatise on PUFA, it seems a waste not to use it. Ignore if you're happy with elongase and desaturase enzymes.
Omega counted double bonds are very straight forward, they are counted from the methyl end of a long chain fatty acid. Because mammals can't add extra chain length or desaturate at the methyl end, these bonds are "fixed" in their identity. So mammals can pop a double bond in to stearic acid in the omega 9 position to give oleic acid and that's it as far as the methyl end is concerned. Double bonds at the omega 3 and 6 positions are also fixed and come from the diet (mostly, there may be an exception). No chance of elongating at the methyl end, so the 9th/6th/3rd will always be the ninth (or 6th or 3rd) bond down from the omega end of the chain. So oleic acid is an omega 9 fat and all of its derivatives are too. Whatever elongation/desaturation happens, it happens at the carboxyl end. Ditto omega 3s and 6s.
Delta refers to desaturase enzyme's ability to change a single bond to a double bond, extracting two hydrogen atoms in the process. So delta 6 desaturase pops a double bond in to the place of the 6th carbon-carbon bond, counting from the carboxyl group end of a fatty acid. This number is the alpha number, as it's counted from the opposite end to omega number. The desaturases mostly don't care how long the fatty acid is, they just grab the acid end, count six (or seven or nine or five etc) and stick in a double bond.
We definitely have a delta 5 and a delta 6 desaturase. Oddly enough we never put double bonds in to adjacent locations in our fatty acid carbon chains, there is always a gap. The pattern goes double bond, two singles, double, two singles etc, as far as I can see. That is if you want extra double bonds at all.
Fatty acids get elongated. This always happens from the carboxyl end, and always involves adding two extra carbon atoms, using single bonds only. In mammals anyway.
So when the fatty acid with a recently added double bond placed in the alpha 6 position by our delta 6 desaturase gets elongated, that new double bond gets promoted from the alpha 6 position to the alpha 8 position.
To keep the pattern we want two single C-C bonds then the new double. That mean going for the 5th C-C bond using delta 5 desaturase.
This neatly gives us the end product of arachidonic acid from linoleic acid.
We also get to eicosapentaenoic acid (EPA) by the same pathway if we start from the alpha linolenic acid parent. Then a simple elongation and a delta 4 desaturation gives docosahexanoic acid (DHA). If your delta 4 desaturase doesn't work you can try this:
Double elongation w/o desaturation to 24 C chain with first double bond now pushed to the 9 position from the carboxyl end. Delta 6 desaturase places an appropriate double bond at the 6 position, as it always does. This weird fatty acid is then shortened by two c atoms (off of the COOH end of course) to give DHA, the new double bond thus ending up, as it should for DHA, at the 4 position (from the 6 position where it was placed by delta six desaturase).
That last paragraph is from Mary Enig's book "Know Your Fats". The rest is general biochemistry.
There is no arguing with the essentiality of arachidonic acid and probably the same goes for DHA. If we don't get them pre formed in our diet, this is how we make them.