First things first: I've worked in a number of different industries, but always related to software development. I will attempt to explain why for my points, so you can adapt them to your situation.
Also, further reading for those who haven't already from a manager's perspective.
What are they?
Generally, when people talk about one on ones, they mean a semi-formal meeting (almost always in the boss' office) scheduled between you and your boss (or you and each of your direct reports) on a weekly or bi-weekly basis. I've seen ones scheduled less frequently, but I do not recommend it (see below for why).
What are they for?
1. Getting feedback
From the manager side of the table, the far most valuable thing I got from one on ones is feedback on how I'm doing as a manager. What works? What doesn't? Since your boss will infrequently be around to see you interacting with your team, and your team will be wary to provide feedback in group settings - this provides really the only time for you to get that feedback.
More often than not, one on ones end up being bitch-fests (for lack of a better term). As a manager, you will often end up acting as a stand-in therapist. This is useful to you as a manager because you can learn what frustrates your reports and why. You learn what is important to them. You build trust that you're someone your reports can talk to without fear of judgement.
As an employee, having a manager who is capable of listening - and who at least feigns interest in what makes my job annoying/difficult/frustrating is invaluable. It is the single largest thing that contributes to my job satisfaction. I understand that you won't be able to solve all the problems; maybe none of them. But by simply caring about the problems I face is a step above many managers.
3. Giving Feedback
There's an old adage about yearly reviews that "in the best reviews, nothing is a surprise". This is your opportunity to provide your direct report with feedback about how they're doing. What is good? What could they do better? How? What did you discuss at the last 1:1, and did it work?
4. Learning to work together as a manager and employee
This ties into feedback, but is important enough to get its own bullet. Everyone is different. Managers have different management styles. Employees have different work styles. People have different personalities. The relationship between a manager and their direct report is a special one though. Once you get that relationship where one person can tell another what to do (or else), things get weird. The one on one is the time for the two of you to work out the differences that make you each special, and line it up with the whole "order you around" thing.
The manager needs their direct report to do well for them to do well. And the direct report needs their manager to do well for them to do well. Now you need to figure out how you can best help each other do that.
5. Career Planning
Along those lines, the one on one is the time to talk about all of the long term things that matter to the direct report. Where is my career going? How am I going to get there? Too often people are swamped in the day to day work to focus on the long term work to make themselves the best employee they can be. As a manager, this is where you can save yourself a lot of headache later by helping your reports be satisfied with their career, not just their work.
6. Clarifying/Reinforcing Messages
How many times have you seen a presentation, and it concludes with "okay, any questions?". All the time. How many times have there actually been questions? It's uncommon. As a manager, here's your opportunity to follow up with your direct reports to make sure that they understand the message given, and have a private opportunity to voice concerns or ask questions that due to human nature, were left unasked in the group.
What are they NOT for?
The number one thing that managers can do to harm their team is to cancel or be consistently late to their one on ones. It's essentially saying to an employee "your well-being is not as important to me as X". A quick and effective way to kill employee engagement.
2. Talking about yourself
As an employee, I don't want to have a meeting where I hear about the boss' Schnauzer. As a manager, I actively don't want to hear about your wife/kids lest I be accused of discrimination later. Most importantly though, socializing isn't something that's limited to the boss and the direct report. "Get to know you" time is best served with the whole team, so everyone can get to know each other.
This is an odd one I know. Praise is a necessary part of feedback, but if the direct report is truly deserving of praise, then they are deserving of public praise. In today's workplace, where teams so often interact having a manager come out and say "Bob did a great job with X. Thanks Bob!" is powerful.
You don't need to meet in person to get status. As a manager, if you need to wait a week or two to get status, you likely don't know what's going on.
4b. Current Projects
Along those lines, a one on one isn't really about solving problems with your current work. Sure, some things will come up, but you spend the other 39-79 hours a week doing work. Spend the time on employee improvement. Also, any current work problem should be dealt with as a team, not one on one.