In general, the term comes from the idea that if the person's door is open, you're welcome to pop in, say hi and bring up anything relevant to the work or your mutual relationship. It's a way of saying "I won't be bothered if you interrupt me when my door is open". When an executive or upper level person who is several ranks of management higher than you says it to you, it also implies that you shouldn't worry too much about the power distance between yourself and this person, you should feel free to interrupt them anyway.
It's a general contrast to the old school formal way of relating to management by booking a meeting or having a formal invite to talk.
Differences in Relationships
Keep in mind that the general rules of office politics are not suspended in this situation. You're still communicating within the office context. That can be highly variable. Some offices are very casual, some are formal. Some have "high power distance" relationships (be very respectful and deferential to powerful people), some have "low power distance" relationships - where you can debate, argue, discuss and strongly question management.
When it's your direct supervisor, you can generally have a long detailed talk about the work, the issues and how to solve them. Hopefully any relationship with your management involves you and your boss listening to each other sorting out how best to relate to each other and get the job done. There may still be a formality to the relationship, but it's essential to have a good working relationship with your direct supervisor and that generally takes time.
For most other types of people with an open door policy, these tips usually apply:
Ask if they are busy before popping in. People forget to close their doors in a moment of deep concentration, and they may want you to hold on for just a minute or two.
On first contact - keep it short and sweet - introduce yourself, say something positive, focus on anything that ties to your shared connection.
Make sure that you work the problems and criticism of current activities in a proper way. No boss likes to hear that you went to another person first with an issue that relates to his team. OTOH - if your boss says "hey, check with X on that" and X has an open door policy - then just head on over with that issue!
Be respectful of a person's time - if you have an issue that you want to discuss that will take more than 10 minutes, book a meeting with the person rather than waiting for their door to be open.
Find a balance in frequency - the more highly ranked the person, the less frequently you want to pop in and say "Hi!". One trick I've used is that I'd only pop in to say "hi" to the VP once a quarter, because there was usually a state of the business speech he'd presented and I could ask a meaningful question regarding the content. If I had not meaningful question in mind, I might not even pop in.
Watch for cues that your timing is off. If the other person is distracted or agitated, say a quick goodbye and come back later.
All sorts of people will have open door policies. Some do it because they read in a book that it was a good practice. Others do it because they genuinely like connecting with people across all positions in the company. Some types of roles almost necessitate it - HR and legal may be the most likely to have this policy in the hopes of encouraging employees to come forward when there are related issues.