Like everyone else has said, it depends on your advisor and his/her personality and style, and same goes with all other faculty.
My department's pretty diverse in how it goes. A few faculty always prefer the formal "Dr." or "Professor," some are always on a first name basis (with a few obvious exceptions in some professional settings), and some are in-between, depending on context of the situation.... and it also depends on your standing with them too. I can imagine it being disrespectful for a first year student to address some faculty by their first name (nickname), while it being O.K. for more senior grad students to do it. Personally, I'm comfortable with casual conversation, cracking jokes in the break room, etc. with just about all the faculty in me department. A handful of them I address by first name most of the time, the rest I still greet "Dr. X", but this is mostly with faculty I haven't worked with or talked to much. General rule of thumb for me is always be formal until they indicate that you can address them by their first name. Generally they'll tell you if that's what they prefer, or the way they sign their emails is a good indication.
We have department happy hours where grad students, faculty, staff, etc. all hang out and drink a few beers and it's perfectly acceptable for grad students to be there, but in actuality only a handful of grad students (myself included) actually go.... but we (the grad students that do go) tend to be the ones on a more casual terms with faculty. I think this is an important thing to do actually... it can be greatly beneficial when it comes to professional recommendations and collaborations (well, assuming you're not a jackass in your outside life). Actually, I've sat at a hotel bar after a day at the conference and had 3-4 beers with a few professors before, sharing stories, jokes, etc.
With regards to my mentor/advisor, sometimes a good bit of our weekly meeting is spent making jokes and talking about random stuff. In our meetings, it's not even a surprise to drop a few curse words... I'm pretty sure I've even had a smiley face sent in an email from my mentor before, lol.
tl;dr: It depends on the person and your standing with them