I've always felt a strong advisor/advisee relationship is the single most important factor in a successful PhD career. At its best, the advisor works closely with the student to successful research agenda and help mentor them through their graduate career and beyond. The advisor/advisee relationship can feel like a parent/child relationship that lasts an entire career. Nothing gives me more pleasure as an academic than to see the success of my current and former students.
Take your time when picking an advisor. Don't choose an advisor based solely on research area or because they are "famous". Pick the advisor that will best guide you to a successful academic career.
At its worst, a bad advisor/advisee relationship will destroy your graduate career, making you feel miserable, perhaps dropping out of graduate school or worse, particularly if a student doesn't feel like they are being treated fairly.
Two incidents prompted this post. On TCS-Stack Exchange, a student has authorship issues with their advisor. Unfortunately these kinds of incidents happen more often than one suspects. If you can't work it out with the advisor, go talk to someone about it, another faculty, the graduate or department chair, a grad student ombudsperson if your institution has one. We care about our students, and will work hard to resolve problems.
In a much more tragic event, a student felt it easier to take his own life than feeling that he had to cover up potential academic misconduct. Again, if you ever find yourself in such a situation please reach out. Giving up is never the answer.