Before I start thinking more about the details of how to practise hospitality etc, I thought it might be good to do a short post saying what I think it means.
In Greek, the word hospitality (Philoxenia) was originally made from two words, a word that means 'love' and a word that means 'stranger'. However, I don't think it needs to mean only strangers simply because xenos forms part of the root of it. (We don't assume that every time someone says "Goodbye" they mean "God be with you"!). 'Hospitality' and related words are sometimes used in contexts where it's clearly not strangers. For example, 1 Pet 4:9 talks about offering hospitality to one another without grumbling, Rom 16:23 - Gaius is "xenos [here the word is used to mean 'host'] to me and the whole church".
There is a strong connection between hospitality and love - not just the fact that the Greek word for 'love' is part of the word, but also the way in which hospitality seems to be spoken of in context as a concrete expression of love - eg. 1 Pet 4:8-9, Rom 12:9-13, Heb 13:1-2. This suggests that hospitality is about being other-person centred. It is not about showing off how well you can cook, or how clean your house is, but about welcoming people and meeting their needs above your own.
It seems from the passages I mentioned the other day, that welcoming people in is a big part of hospitality. A friend of ours uses the very helpful illustration of seeing hospitality as (temporarily!) expanding the boundaries of your home and family. This will mean in most cases that it will involve having people into your actual house, including them within your life - catching up for coffee or dropping a meal at someone's house are good things to do, but I'm not sure they're hospitality.
You can find a more detailed discussion of what hospitality means in the Bible in Baker's Evangelical Dictionary of Biblical Theology.