Blessed are they that mourn: for they shall be comforted. (Matthew 5:4)

Is this a promise to all who mourn, and for whatever reason they mourn, or are there any terms or conditions attached to this promise of the Lord? Must the mourning be over a certain subject? Must the mourner belong to Christ?

The people of the Lord are comforted in their understanding of the purpose of life, suffering and death. So when they become mourners due to suffering and death they will more readily be comforted.

The Holy Scriptures are filled with examples of times when mourning was a punishment for sin. People who mourn that they are being punished for their sins, rather than that they have committed them in the first place, will not be comforted, until their mourning is right before the Lord. People who mourn their own failure to obey the commandments of God, and who mourn that they have hurt or injured other people whom they should rightly have blessed; as they turn unto the Lord, with a broken heart and contrite spirit, they will be comforted.

God's agent in comforting those who mourn is the Holy Spirit (or the Holy Ghost, or the Spirit of Truth, for they are all the same at this instance).

26 But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall testify of me: (John 15:26)

Since the Spirit of Truth knows all things, and since he at the exact right time can dispense the exact right understanding to the mourning heart, he is truly a comforter.

16 And I will pray the Father, and he shall give you another Comforter, that he may abide with you for ever;

17 Even the Spirit of truth; whom the world cannot receive, because it seeth him not, neither knoweth him: but ye know him; for he dwelleth with you, and shall be in you.

18 I will not leave you comfortless: I will come to you. (John 14:16-18)

The Spirit of Truth is a gift that is given by the laying on of hands. So, real comfort to mourners who are true followers of Christ, as promised by him on the mount, requires Baptism by immersion, and having received the Holy Ghost by the laying on of hands. Further, If our mourning stems from our own inabilities, a sacrifice of a broken heart is also essential. But the comfort that is given by the Holy Ghost is wonderful, and well worth any subjection on our part.

The Gift of the Holy Ghost is not a part of the ministry of the Aaronic Priesthood. John the Baptist made it clear that he would not give the Holy Ghost to those whom he baptized. He told his followers to go to "one who is mightier than I… He shall baptize with fire and the Holy Ghost". This means that before Jesus came with the Higher Priesthood, the Israelites did not have the comforter. Since they could not take the full gospel they were not given the full comfort and peace.

1 SO I returned, and considered all the oppressions that are done under the sun: and behold the tears of  such as were oppressed, and they had no comforter; and on the side of their oppressors there was power; but they had no comforter. (Ecclesiastes 4:1)

Jesus himself is another Comforter who will come to meet the very elect upon the Mountain of the Lord.