Why can we say 'an American' but not 'a British'?

I am confused with the use of an indefinite article in front of British or Chinese. To my understanding, we can place an indefinite article in front of any “countable noun”. So, we can say a cup and an orange.

But when it comes to nationalities, it is very confusing. For example, we can say an American or a German. But we cannot say a British or a Chinese.

I looked them up in a dictionary as at first I thought British and Chinese are adjectives in the above statement and that is why we cannot place an article in front of it.

However, the dictionary mentions that they are nouns. Does that mean they are uncountable nouns?

The noun for an inhabitant of Britain is Briton.
British is an adjective.

For many countries, the adjective and noun are identical. As you've found, German and American are good examples.

The noun for an inhabitant of China has historically been Chinaman but in recent times, the word Chinese has been increasingly used.

Your dictionary may list British and Chinese as nouns because they are used as collective nouns, to refer to the population of each country as a whole or as a generalization:

The British are mad about football (soccer).

Compare with

Americans are obsessed with football (not soccer).

The British is really just a short form of The British people, with the noun presumed. In the same way, you can say "I'll have the red" when someone asks you what type of wine you want -- the noun (wine) is presumed, so the adjective stands on its own.

The problem is that sometimes there is an easily available adjective form but not an easily available noun form.

This is also complicated by the fact that we can form a collective noun from an adjective by adding the definite article, such as the rich or the famous. Now British, English and Welsh are adjectives, and we can certainly talk about the British, the English and the Welsh. We cannot however say a British, an English or a Welsh because the proper noun forms are different. We need to say a Briton (or a Brit, as noted by Erik), an Englishman or an Englishwoman and a Welshman or a Welshwoman.

There are other instances when the adjective form and the noun form are identical: you have already noted American and German. Other examples are Italian, African, Israeli, Indonesian, Pakistani, Indian and Brazilian. You might note that many in this category end with -an or -i. This might be useful as a rule of thumb for adjectives that can be used as nouns too.

I upvoted @Chanmunka's answer, but let me add an additional thought that's too long to fit in a comment:

Yes, the usage is inconsistent, or at least appears inconsistent.

For some nationalities, the noun for a person from that place and the adjective for that place are the same. "Bob is an American. Bob drives an American car." "Erwin is a German. Erwin drives a German car."

For others the words are different. "Richard is a Briton." But, "Richard drives a British car."

Oh, and by the way, often you make the adjective by dropping any final vowel and adding -an to the noun. Like America -> American, Mexico -> Mexican, Venezuela -> Venezuelan, Russia -> Russian. If it ends in -y you change the -y to an -i-: Italy -> Italian. But for some reason Canada does not become "Canadan" but Canadian. Other times we drop the final syllable and add -ish. Like Britain -> British, Poland -> Polish, Ireland -> Irish. And then there are always the weird cases, like Netherlands becomes not "Netherlandan" or "Netherish" but Dutch.

For many nations the convention used to be that we used the adjective plus -man or the name of the place plus -man to refer to a person from that country. "Francois is a Frenchman." "Li is a Chinaman." Etc. As the conversation on another post indicated, some now consider that sexist, but no clear alternative has become accepted, so we're rather stuck using a phrase, "Francois is from France" or "Francois is a French person."

When the adjective is different from the noun, you can use "the" plus the adjective to describe the people of that nation as a whole. "The British like to drink tea." "The Chinese built the Great Wall." When the adjective is the same as the noun, you use the plural to describe that nation as a whole. "Germans are hard-working people." "Italians are crazy." Etc.