I regret my choice of grad school, is it possible/common to accept an offer I turned down a year ago?
There's almost a 100% certainty that you'll have to apply again. Graduate programs get a fixed number of admission slots (or fellowships to support those admits) each year. Your position was likely given away to someone or returned to the graduate school when you told them you weren't going.

As Nate Eldridge notes, you were a good candidate last year and you'll be a good candidate this year. There's a slight stain on your record regarding whether or not it was a personality conflict or something else that is causing you to abandon the other school. 

If you have a good relationship with your current advisor, then s/he can address that in their letter of recommendation. Or you can ask your undergraduate advisors to note the reason why you're leaving the university.  The faculty on the admissions committee will be curious why you're leaving a more highly ranked program and admissions committees are risk-adverse, so you do need to address this somehow.