How you should approach this depends on what you've been promised and who promised it to you, but usually, the best thing to do is just give that person a gentle reminder.
Favours
If somebody's doing you a favour, you have to respect that it might take time, but if you've been waiting a long time, you might want to ask them again if they intend on following through. Obviously, you'll want to be a touch more polite than that, but they need to know that you need a definitive answer here.
If they're going to help you, you need to remind them how urgently you need their help, and how grateful you are. It's likely that this thing has just slipped their mind, so a gentle nudge won't hurt.
However, if something has come up and this person can't help you anymore, you need to know so that you can find someone who can. Whatever the result, don't get angry at them. Yes, you should do something if you say you're going to do it, but at the same time, you're not entitled to their voluntary help, so you don't have the right to get annoyed at them.
Things You've Paid For, Or Are Entitled To
You can be a bit more forceful in situations where you've paid someone or are owed something, but you still need to talk to whoever it is who made this promise to you.
There's usually no need to approach a subject like this with anger or impatience, and doing so will only make the other person more reluctant to help you. Start off with a gentle reminder, and then another one a few days (or weeks, depending on how big a favour this is) later.
If this doesn't work, tell the other person that you need them to follow through with their promise urgently, and that, if you'd known they were going to take so long, you could have found somebody else to help you more quickly. There's no point saying this if the situation is out of the other person's control, though, or if you couldn't have got anybody else to help!
Be Sympathetic: Consider Their Situation
Remember that other people have their own things to worry about, and that there's probably a good reason that this person hasn't been able to provide you with what you need.
If somebody doesn't have much money or time, you shouldn't have demanded it of them in the first place, and this could be why they've not been able to help you yet. Sometimes, people say they'll do you a favour — or even volunteer — because they want to please you, and don't realise how difficult or inconvenient it will be to follow through with.
The most important things to do in any situation like this are to get your point across, be polite, and be understanding. If this doesn't work, look elsewhere for help.