North America
The most common way of classifying wheat flour in North America are terms such as whole wheat flour, bread flour, all-purpose flour, pastry flour, etc. This is a simplified system designed to indicate the best use of each flour. Want to make a loaf of bread? Use bread flour. Making pastry? Use pastry flour. Want a flour that you can use for a variety of things? Grab some all-purpose. And on the surface this seems pretty straightforward, but it's important to note that these are essentially marketing terms. While there are regulatory bodies that govern classification, there are no strict definitions to differentiate between say all-purpose flour and bread flour.
What you have instead are general ranges that are mostly tied to protein content. And while total protein can be an indicator of potential gluten strength, there are other factors which also determine a flour's suitability for a specific application. For example, there are all-purpose flours milled from soft wheat which are not suitable for bread making, but there are also some all-purpose flours that are actually stronger than some bread flours. In fact, the same exact flour might be labeled as all-purpose by one company, and as bread flour by another.
So what does this mean in terms of choosing the flour that's right for you? It means that in order to make that determination, there are more factors to consider. Let’s go over the different types of wheat, along with other factors that determine flour suitability.
United Kingdom & Ireland
Here you have a similar system to North America, but with different names. You'll see names like strong or strong white, or even extra strong or very strong. These are roughly equivalent to the bread flour designation across the pond. Some say that plain flour is roughly equivalent to all-purpose, but in many cases an all-purpose flour in the US like King Arthur would be more equivalent to strong white flour in the UK. A lot of millers in the UK now import grain from North America (primarily Canada) and use this grain entirely or as a blend to make stronger flour than could be made from native British wheat alone.
Again, this is a highly subjective system, and with the potential for different types of wheat to be blended into a single flour, it may be harder to determine the baking characteristics based on naming alone. Another wrinkle in the whole flour equation is the fact that US and European methods for measuring protein content are different. The European method measures protein based on dry flour content only, while the US measures is based on 14% moisture in the flour. So if an American flour and European flour have a protein content listed on the label as 10%, the American flour would actually be 11.4% protein if measured in Europe.
France
The system of classifying wheat flour in France is numerical, and rather than loosely being related to protein content, it is instead tied to ash content of the flour. Ash content in flour refers to how much of the bran and germ remain after milling and sifting it. What is actually being measured is the mineral content, but the way that is measured is by incinerating a portion of the flour, then weighing the resulting ash that is left behind. Since the trace minerals present in flour come from both the bran and germ fractions, the higher the ash content, the higher the proportion of bran and germ present in the flour.
Examples of French flour types are T45, T55, T65, etc., with the number designating that percentage of ash. T45 has an ash percentage of around 0.45%, whereas T55 is approximately 0.55%, and so on. These designations roughly point to an estimate of strength related to protein, but also the effect that bran has on the ability of gluten to form effectively. In my opinion, this system is generally more accurate than the British or American ones in terms of determining flour suitability for specific application, but it is still lacking in how a flour might perform specifically once made into dough.
Germany
The German system is nearly identical to the French, but the numerical values are greater by a factor of 10. For example, a T55 French flour in Germany would be classified as 550 flour. Like the system in France, it is based upon the ash, or mineral, content of the flour. The German system also extensively classifies rye as this grain is used extensively, either blended with wheat or on its own. The rye classification, like wheat, is based on the extraction level, which indicates how much of the whole grain is present in the flour.
Italy
The Italian system for flour classification is the most specific of all. They have numerical types of flour which range from 2 to 00, but they have also adopted a system which gives numerical data for how the flour specifically performs as a dough. The Tipo classification measures ash content, with the finer grades containing progressively less, and additional data are often expressed as a W value and P/L value. These data are obtained by subjecting the dough to a mechanical stress test using a Chopin Alveograph machine. This results in a W factor number, which expresses the flour's strength, or elasticity, and P/L describes the balance between dough tenacity and extensibility. Together, these numbers accurately describe both the performance of the dough under real-world conditions, and the suitability of it for specific applications.