Htydjtk

What are the regulations for relationships on and off the clock?

In the new generation (Star Trek XI) Uhura kisses Spock in front of Captain Kirk. Now , Kirk and Spock were just about to try and board Captain Nero's ship to save Captain Pike. So it can be safely assumed that at least Spock was on duty as he and Uhura kiss for a while in front of other crew members and Captain Kirk.

Now I know that moments earlier, when Captain Kirk says he is also coming to try and save Captain Pike, Spock says to him:

I would cite regulations, but I know you will simply ignore it...

So no-one would expect Captain Kirk to make an issue out this, as he is not one who follows Starfleet regulations to the last word.

Was there any information on crew members fraternizing or having relationships? And what were the rules when a crew member was on duty?

It's the wrong timeline and 80 years too late, but it's pretty clear that in Star Trek: The Next Generation there is nothing resembling a ban on fraternization. Several major characters — including Data, who would certainly cite such a regulation if it ever became relevant — have romantic relationships with shipmates over the course of the series without worrying about violating such a ban.

The one time the idea of a fraternization ban comes up is in "Lessons" (about a romance between Picard and one of his department heads), which explicitly denies its existence:

PICARD: Sit down, Counsellor. I want to talk to you about a matter of protocol. I know there are no Starfleet regulations about a Captain becoming involved with a fellow officer, but —

TROI: You would like my opinion about you and Commander Daren.

In the Kelvin timeline, the only evidence that I know of is Spock and Uhura's relationship itself; again, it seems unlikely that they would be romantically involved if they knew of some regulation forbidding it.

2150s

Starfleet apparently had a restrictive policy during this time period. It would seem that having sex with someone outside of your chain of command (and of the same rank) is acceptable, but otherwise not.

ARCHER: We heard you. What makes you think we're suffering from a lack of sexual activity?

TPOL: Starfleet forbids officers from fraternising with subordinates. Unless you've been violating regulations

_{^{Ent: Fallen Hero}}

2260s (prime timeline)

In The Original Series, there were apparently no specific rules that forbid a Starfleet officer from "fraternising" a non-Starfleet person.

KIRK: Lieutenant McGivers' idea to welcome Khan to our century. Just how strongly is she attracted to him?

MCCOY: Well, there aren't any regulations against romance, Jim.

_{^{TOS: Space Seed}}

2360s

In Voy: Scientific Method we get to see precisely what the rules are regarding 'fraternisation' both on the clock and off. In short, you're welcome to pursue a sexual relationship with whoever you want, but do it on your own time, and do it discreetly.

JANEWAY: Tuvok? I haven't heard a word from Tuvok. You two have been making enough of a public display that half the ship is gossiping about it.

TORRES: Believe me, that wasn't our intention.

JANEWAY: You are senior officers and I expect you to maintain the standard for the rest of the crew, but this adolescent behaviour makes me question my faith in you both. If you choose to pursue a relationship that's your business. But you consider yourselves under orders to use better judgment about it. Is that understood?

PARIS: Yes, ma'am.

To add to Micah's answer, in the original series of Star Trek during the episode 'Balance of Terror', where the Enterprise came under attack by the Romulans, Kirk was due to officiate the wedding of two of the crew.

http://memory-alpha.wikia.com/wiki/Balance_of_Terror_(episode)

In the Star Trek: Voyager episode Elogium, Captain Janeway implies that there are no Starfleet regulations against fraternization off the clock:

CHAKOTAY: This morning I interrupted a couple who were kissing in the turbolift, and I've been wondering if we should establish a policy regarding fraternisation.

JANEWAY: Well, the couple in question might be urged to show a bit more discretion, but Starfleet has always been reluctant to regulate peoples' personal lives.