Ahlan wa sahlan is the phrase you’ll hear most often in Jordan. It’s most commonly rendered as “welcome”, but translates directly as “family and ease”, and so might come out better in English along the lines of “Relax and make yourself at home [in my house/shop/city/country]”. With hospitality a fundamental part of Arab culture, there’s no warmer or more open-hearted phrase in the language. Everybody uses it, in all situations of meeting and greeting, often repeated like a mantra in long strings.
As a visitor, you needn’t ever say ahlan wa sahlan yourself, but you’ll have to field torrents of them from the locals. The proper response – even if you’re walking past without stopping – is ahlan beek (beeki if you’re talking to a woman). Alternatively you can acknowledge the welcome with a smile and shukran (“thank you”) or an informal ahlayn! (“double ahlan back to you!”).
The catch-all word used to invite someone – whether welcoming an old friend into your home or inviting a stranger to share your lunch (surprisingly common) – is itfuddal, often said together with ahlan wa sahlan. Translations of itfuddal (itfuddalee to a woman, itfuddaloo to more than one person) can vary, depending on circumstance, from “Come in” to “Go ahead” to “Can I help you?” to “Here you are”, and many more. A respectful response, whether or not you want to take up the offer, is to smile and say shukran (“thank you”).