Dr. Monzer Kahf, a prominent Muslim economist and counselor, states the following: Firstly, Al-Hajah (need, not a necessity) that affects large group of people or the whole community: Al-Hajah is a degree less than necessity. Necessity is a thing without which damage may occur to one of the five things for which the Shari’ah caters most. These are, in sequence: religion, life, intellect, lineage and property. Its example is mentioned in the Qur’an (16: 106). A person threatened with death, and he knew it was serious because his father had been killed for the same reason, and forced to deny faith and curse Prophet Muhammad (peace and blessings be upon him).
On the other hand, lack of al-hajah does not hurt, but it makes the achievement of the objective difficult in a way that creates hardship for people. Al-Hajah fulfills an objective less than a necessity but more than just making things simply better. It is something that is essentially better, in the sense that without it, the fulfillment of the objective of the Shari’ah becomes substantially difficult. Keeping in mind that every obligation in the Shari’ah (as in any other system except anarchy, notice here we are discussing in terms of pure human rationale) implies a certain cost or effort, al-hajah is beyond that level of cost or effort; it is a matter of hardship or a very crucial benefit.
Al-hajah (and the same applies to necessities, but to a lesser extent because necessities are more extreme) can only be determined or assessed by the concerned person himself/herself, no body else. He/she knows for sure his/her needs, he/she knows what creates hardship for him or her, what is essential and what is not. One must realize that one is dealing with Allah Who knows best and is aware of everything, i.e., things that we keep in our hearts and things that we reveal. The assessment of hajat (plural of al-hajah) differs from one person to another; this is correct and natural. For instance, a person gives more value to security than he does to money, while another sees it the other way round.
The second part of your question is about what hajat are as regards the issue of house financing under the American law.
Hajat that are mentioned in the Fatwa on house financing are of the following kind, examples only not inclusive: the need to have suitable neighborhood to live in, decency, non-involvement in drugs and crime, good schools, availability of rental housing with comparative cost, availability of housing for large family, closeness to Muslim and/or non-Muslim peers, the tax effect and the final cost, building equity, etc.