It seems to me that the proper way to refer to a time period in history is to refer to the century… am I right about that? Are historians, writers etc taught that?
Is there a good reason for it? Personally, I have to convert xx century in my head for it to register.
I don’t know which is the “proper” way to refer to an entire century. I’m not sure it is something that’s taught to historians; it is probably more a matter of personal taste and whatever writing style you’re trying to follow.
That said, I think there is a distinction. “19th century” refers to the years 1801-1900 inclusive. “1800s” would refer to 1800-1899. It is a subtle difference, but different nonetheless.
Historians by and large use “XX century,” perhaps because it sounds slightly snobby. There isn’t a “proper” way, but we do have something of a convention.
I tend to refer to “19th century” and “20th century” because I do deal with both those periods when researching building histories here, and they are a comparison with each other, as much as they describe when the buildings were constructed. “1900s” tends to be in connection with the word “early”, when referring to the first few years of the 20th century.
Not really a convention – just seems to make more sense that way.
Oh, and definitely no apostrophe in “1800s” or “1900s”. Simply not done, ol’ bean! 
I prefer using ‘the 1800s’
1800s ought to mean the first decade of the 19th century. Followed by the…uh…18teens.
Now I’m curious, what is the range in years for the 1st Century?
No, no. 18tens (analogous to 18twenties, 18thirties, etc.). The teens are 13-19.
A.D. 1 through A.D. 100. There was no year “0”, only a 0 point — 1 B.C. led directly to A.D. 1.
Not that I’ve had much chance to use the construction, but I’ve been thinking about this: how does one distinguish between “the 1800s” meaning “the entire century whose numbers start with 18” and “the years 1800 through 1809”? If I say “the 1800s”, I think “the entire century”.
I hereby propose that the first decade of the 1800s, the years 1800 through 1809, be called “the eighteen-zeros”. For the entire century, we would say, “the eighteen hundreds” as usual.
Thus, we are now living in the decade of the “twenty-zeros”. As well as the century of the “twenty-hundreds”. And, come to think of it, the millennium of the “two thousands”.
It occurs to me that it would still be written “the 1800s” or “the 2000s”, though. For clarity, you’d need to write out the words.