I have been reading books recently. In some books, the dialouge looks like 'Hello, Veet.' but in some, it looks like "Hello, Veet." Is it the change of American or British English? Or is it just the author's mistake?
It's standard in Britain to use a single quote--'. American authors will use double quotes--". It doesn't really matter which you use, just be consistent.
In school we were taught to use the double speechmarks (British school), so I've always done it that way. I don't know if it's just because it's a better visual aide while commas and apostrophes can be confusingly the same shape, so using the double ones makes for better learning material. In any case, I automatically type the double quotes, so if anyone were to tell me I had to write using single for some pedantic reason (or, more reasonably, because I had a story going into a collection where most everyone else was using single quotes), I'd write it normally, then do find and replace.
Schooled in England in the 1950s/1960s and was also taught this " was a speech mark. Melzaar the Almighty, much younger than me was taught the same. So, why do I keep reading on grammar websites that the Brits do it differently to the USA?
Because we can do it ' and just as many of us taught in the modern age of computers were taught '. A Brit can do either an American doesn't seem to have the same freedom.
'Standard' is too strong a term. The Oxford Style Manual says that single quote marks are 'normally' used for outer quotes and double quote marks for quotes-in-quotes, but observes that newspapers often reverse that convention. I was taught to use double quotes for long passages and single quotes for single words or short phrases. In other words, in British usage there is no standard for whether outer quote marks should be single or double, just lots of contradictory house styles. Finding one convention in one place and another convention in another publication doesn't strike us as a "mistake" because we never have any expectation of consistency between publications.
Double quotes are the American standard. In the UK and many British Empire nations, the tradition is single quote, but modern usage appears to be dhifting in facvor of double quotes. This applies to the outermost level of quotes. If there is an inner quote, it uses the opposite form. If you have inner-inner quotes, you should consider rewriting the passage anyway.