It is just the root of the coriander plant.
Certainly at least here (UK) you can buy living coriander plants in the supermarket; you could pull one out of the pot and use the root from that.
Apparently you can also subsitute 2 stems of coriander for every piece of root called for in the recipe, but I've never tried this.
cooking.stackexchange.com/users/197/Vicky
In the USA, Coriander is referred to Cilantro, when used in context of herb/green.
Possible sources (to purchase Cilantro aka Coriander with root): vendors at some farmer's markets
(this is where I get mine)
"South East Asian" or "Latin American" grocery stores "Indian" Grocery stores
cooking.stackexchange.com/users/470/joyjit
If it's not carried in a local Asian market (and they won't order it) then try asking at a local Thai or Asian restaurant.
If they're using it they might very likely be willing to sell to you or order from their supplier.
cooking.stackexchange.com/users/426/Darin Sehnert
The following is a quote from the Blue Dragon website , I hope it helps:-
Coriander roots are notoriously difficult to come by (even in well stocked Oriental supermarkets), as in the UK coriander is sold rootless.
This isn't much good for budding Thai cooks!!
To get the same intense coriander flavour you would get from one root, use ten stems stripped of the leaves and pound to a paste in a paste and mortar.
Use in the same way as you would have done with the root.
N.B.
Many supermarkets in the UK stock live coriander plants growing in small pots but as previously observed the roots are to small to use.
cooking.stackexchange.com/users/34186/Martindp2
"Cilantro" is actually a stage of growth of the Coriander plant; it is when the leaves are broad and light green, prior to flowering (aka Bolting).
If the plant has flowered the taste becomes increasingly bitter and should not be used in recipes calling for coriander.
I would second the suggestion to buy a live plant and simply take the root.
I've also been told that you can use the stems as a substitute but haven't tried it.
cooking.stackexchange.com/users/552/Gerbil
The only place I have seen coriander roots thick enough to be used in cooking is in Thailand.
The coriander we find in Europe has roots so thin you can't use them.
The roots are much more fragrant than the leaves and yield much better flavouring results when used for cooking.
cooking.stackexchange.com/users/16844/Adeline
According to Thai Supermarket Online - http://importfood.com/ - 'A fine quality coriander seed can been used in various curry pastes and other condiment recipes as an excellent substitute for corriander root'.
cooking.stackexchange.com/users/8747/Boonnoon
The best way of getting coriander root is to grow your own coriander on a sunny windowsill; don't bother trying to use it for your cooking as it'll go from scrawny and unable to offer up any leaves, to suddenly it's bolted if you turn your back.
Just leave it, you'll then get seeds that you can dry and great roots.
I've grown the Calypso variety from seed, because it claimed to not bolt (didn't have any luck with that), and got really good sized roots.
You do need to leave it to really get good and beefy.
You also need to take ages to clean them as the compost gets into every nook and cranny, but if you're wanting coriander root it's worth it.
And I personally just froze it afterwards.
cooking.stackexchange.com/users/83378/Talitha Brown
Traditional Thailand we use coriander root as a flavoring.
You can use stems and some leaves for substitute or you can use MSG.
cooking.stackexchange.com/users/24488/Mayura
I mentioned the book Thai Food in an answer to another question. This reminded me of one of my longest running quandaries with it. Many of the recipes mention “coriander root”. I am aware of being able to buy the seeds (whole or ground) or the leaves, but I have never seen for sale anything calling itself coriander root.  Has anyone managed to source this, or am I missing something obvious?
What is coriander root and where can you get it?
cooking.stackexchange.com
<herbs><shopping><thai-cuisine>
cooking.stackexchange.com/questions/2610
cooking.stackexchange.com/users/1230/Jeremy French
Coriander is usually sold without the root in UK supermarkets, but it may be available in Oriental supermarkets or at a local Thai or Asian restaurant. You could grow it yourself or use stalks, but more stalks will be needed as their flavour is not as strong. Coriander root is very difficult to buy. You can either grow your own or substitute it will a larger amount of stems and leaves.
It is very difficult to buy coriander root. The best way is to grow your own plant and then use the root. Instead of coriander root, you can use 10 stems and some leaves.