I would avoid getting into personal details.
This will heavily depend on the company culture, the size of the company, as well as your relationship with your boss and team-mates, however generally speaking people are not interested in dealing with someone who is emotionally compromised. There are two good reasons to conceal your situation:
Admitting that you don't feel up to doing your work because of a relationship problem might label you as a flaky employee, or someone who should not be promoted because they can't handle their emotions.
Sadly not everyone will agree that your relationship problems constitute a good reason to take off time (I am not in this camp, simply pointing it out). If you really want time off, but your boss does not feel it is a good reason, he might give it to you begrudgingly, but again, label you as flaky or unreliable.
I've witnessed someone go through this exact thing (this is in Canada, so it need not apply to your own situation): her husband of 10+ years left her, and she was left absolutely devastated. The pain obviously carried over to the workplace, and after a few emotional displays, she was placed on leave. The office gossip was not kind, let me tell you. When she came back, she was visibly emotionally shaky. This person got some sympathy for a little while, but then many people turned against her: they are there to get a job done and go home to their own problems and family, they didn't want to deal with someone else's drama. It ended with her quitting, and I don't blame her.
It's a sad aspect of our fast paced work environments, and the typically transient nature of the workforce that many people simply don't want to get too involved with their coworker's problems and life - and they certainly shouldn't have to if they don't want to, however we are losing that human touch with one another.
It sounds to me like you already have some reservations about revealing the true reasons for your request to your boss.
If that's your gut feeling, go with it, and don't tell coworkers anything either. Say that you have a personal emergency, and that you need time off immediately. Maybe make up a reason (such as calling in sick), if you think you can get away with the lie, because later, when it comes out that you have split from your partner, people will immediately know why you were away.
Note: I don't generally advocate lying, but it all depends on how reasonable your boss is. Most people would understand, but some people, or the culture of some companies, are really against making allowances for that sort of "personal problem".