Flour types are very complicated, because there are several different variables and different countries use different systems, so it is hard to establish equivalences.

First, the 0000 and 000 grading system is definitely not used in the English-speaking world. It is essentially a Latin American system. References to these grades in English all seem to be related to Latin America, and the fact that English-speaking people interested in flour have to ask what they are shows that they are not familiar with them. As in the forum query cited by Benjamin:

"can you tell me the US flour designation equivalent to Latin American flour designation categorized as flour type 0, 00, 000, and 0000"
http://allrecipes.com/HowTo/all-about-flour/detail.aspx

And in Alan's reference, again, someone who has moved from Canada to Argentina is asking what "Harina de Trigo 000" and "Harina de Trigo 0000" mean.

Generally speaking, accounts of 0000 and 000 refer to their use:

"La harina 000 se utiliza siempre en la elaboración de panes, ya que su alto contenido de proteínas posibilita la formación de gluten y se consigue un buen leudado sin que las piezas pierdan su forma.
La 0000 es más refinada y más blanca, al tener escasa formación de gluten no es un buen contenedor de gas y los panes pierden forma. Por ese motivo sólo se utiliza en panes de molde y en pastelería, en batido de tortas, hojaldres, etc."
http://www.alimentacion-sana.com.ar/informaciones/Chef/harin...

So 000 is used for bread (and pizzas) and has a higher protein content, and 0000 is whiter and used for pastries.

More technically, they are also officially defined by their ash or mineral content, which is a criterion used in many countries, including France, Germany and Italy, but not systematically in the UK or the US. 0000 has an maximum ash content of 0.65%, and 0000 a maximum of 0.49%, according to the Código Alimentario Argentino
http://www.molinoflorida.com.ar/Pages/Descripcion productos/...

This gives us a basis to compare them with UK and US types.

0000 (<0.49% ash) is equivalent to plain flour:
"The lowest ash content is found in so called plain flour (0,46%)."
http://www.classofoods.com/page1_1.html
Or US pastry flour, with ~0.4% ash content and ~9% protein content
http://en.wikipedia.org/wiki/Flour
Or what is known in the trade as patent white flour:
"Top Patent 0.35 - 0.40% ash content: 11.0-12.0% protein
Uses: - Danishes, sweet doughs, yeast doughnuts and smaller volume breads and buns."
http://www.theartisan.net/flour_classification_of.htm
"The finest flour from the early stage of the milling process. UK term: Patent White"
http://www.dovesfarm.co.uk/about/types-of-flour/european-flo...

000 (<0.65%) is equivalent to bread flour, also known in the UK as strong flour:
"If you want to make artisan bread, you want your bread flour to be: [...]
With a protein content of 10.5 to 12%
With an ash content of .48 to .56"
http://www.farine-mc.com/2009/09/flour-brief-recap.html
Or US all-purpose flour:
"ash: ~0.55%; protein; ~11%: all-purpose flour"
http://en.wikipedia.org/wiki/Flour
"First Baker's 0.50 - 0.55%. ash content: 13.0-13.8% protein
Uses: All purpose strong baker's flour, breads, buns, soft rolls and puff pastry"
http://www.theartisan.net/flour_classification_of.htm

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Note added at 3 hrs (2011-10-22 17:22:35 GMT)
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Sorry, I made an unfortunate typo in the middle of my explanation: 000 has an maximum ash content of 0.65%, and 0000 a maximum of 0.49%

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Note added at 4 hrs (2011-10-22 18:32:46 GMT)
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0000 is equivalent to 405 in Germany, 40/45 in France and 00 in Italy.
000 is equivalent to 550 in Germany, 55 in France and 0 in Italy.
But "in the United States and the United Kingdom, no numbered standardized flour types are defined"
http://en.wikipedia.org/wiki/Flour


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Note added at 4 hrs (2011-10-22 18:36:25 GMT)
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So in the UK, the everyday terms are plain flour (0000) and bread flour (000). In the US they would be pastry flour (0000) and all-purpose flour (000). "Patent" or "patent white" is an alternative trade term for 0000.
In other words, bread flour and all-purpose flour are really the same thing.