This varies greatly between departments. In some cases (particularly European universities, I believe), Ph.D. students are admitted by individual professors. There may be some basic requirements set by the department or university, but other than that it's up to the professor's judgment.
In other cases, decisions are 100% made by a committee. For example, that's the case in my department. Any professor can talk with the committee members and make a case for a particular candidate, but it would be unseemly to do this too often, and the committee can simply ignore these discussions if they choose to. (In practice, they are willing to be convinced if someone makes a compelling case for why the student would be great, but they won't admit someone just because a potential advisor is in favor.)
Many professors say on their websites not to email them because they cannot alone do not make the admission decisions, but rather it is the decision of a whole committee.
That's true in my case. I can't evaluate most applications in isolation (unless someone is mediocre or obviously amazing, I won't know how they compare with the other applicants this year), and I'm not interested in lobbying the committee except in unusual cases where I have important information not available to them. (If they know everything I do, then I don't want to offend them by suggesting that I don't think they are capable of doing a good job.)
However, my undergraduate advisor said there is nothing stopping a single professor from admitting a student if they see fit. Is this true? And if so how often does it happen?
So the answer is that this is completely true in some departments, completely false in others, and may of course be in between in some cases. How often it happens depends on the circumstances (what you are applying for and where). If you are applying to U.S. math grad schools, for example, essentially all the decisions will be made by committees. If you are applying for something else, then you'll have to look into that particular case. But if you are seeing lots of statements on web pages saying that decisions are made by committee, that tells you something.