Is one of these more correct?
I will apply to university next year.
I will apply to a university next year.
I go to university.
I go to a university.
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It's nice to find a usage where the Brits are ahead of the curve for once (I went to university, obviously – as did most UK undergraduates by the mid 60s).
Americans have been slower to adopt the new usage, but they're getting there...
As implied by @M. Tibbits's answer and comments thereto, I think including "a" implies slightly more "exclusivity/cachet/prestige", notwithstanding that many Americans will say "university" means the same as "college" (US speakers go to college, just as in the UK we go to university). I suggest that Brits are following this principle when they speak of going to a redbrick [university].
I believe it is more American (as per FF's answer below) to use 'a' before the common noun 'university'. But if it were replaced with a proper noun, the use of 'a' is incorrect.
I go to Cambridge. I go to a university.
I will apply to Cambridge. I will apply to a university. (only one?)
The ones with "a" are definitely acceptable in American English. The ones without sound British to me... though we'd also say "go to school" and "go to college", "go to university" isn't as common an idiom.
In Australia, I would find "I go to a university" a bit awkward. "I go to university" would be overwhelmingly more common, as would "I'm at university". "I go to a university" sounds like the speaker is making a point that they only attend one one, or that a university is an unusual thing. It almost sounds like they're being enigmatic about which one they are attending, perhaps because it isn't very prestigious.