In some Hadiths, people whom prophets and martyrs will envy are mentioned. What are their characteristics?

According to the narration of Hazrath Omar, Prophet Muhammad (peace be upon him) said:
Surely, some of God blessed servants are neither Prophets nor Martyrs; they are special People, who on the Day of Reckoning, the Prophets and the Martyrs will envy them for their ranks and nearness to their God.

A Companion asked, O Messenger of God (pbuh), who are they, and what kind of deeds do they perform so that we may love Them as well?"

Messenger of God (pbuh) said, They are People who love one another in God, even though they have no consanguineous ties, money to exchange or worldly business to barter with. I swear by God on the Reckoning, their faces will be effulgent lights and they will be raised on Pulpits of Light. They will not be subject to fear when the creation is seized by the awesomeness of the Day of Resurrection, nor will they be subjected with sorrow when the rest of creation will be seized by it." He then recited, Surely, Gods Deputies are not subject to fear (both in this world and the next, for they will always find My help and support with them) nor shall they grieve (2)

Another narration is as follows:
Surely, among the slaves of God, there are the ones whom prophets and martyrs will envy. The companions said, O Messenger of God, who are they; inform us so that we love them. Upon their wish, the Prophet said, They are such a community that although they have no property (transaction) and kinship between themselves, they love each other. Their faces are full of light. They are on pulpits of light. They continue not to fear when the public fears. They do not grieve when the public grieves. The Prophet then recited the verse: Know well that the friends (saintly servants) of God – they will have no fear, nor will they grieve. (2)

The Prophet describes his brothers that will come after him in another hadith: As is narrated by Abu Hurayra, One day Prophet Muhammad (PBUH) came to the graveyard and said: O dwellers of the land of believers! Peace be upon you. With God willing, we will also join you. Then he sighed with longing and added, How I miss to see my brothers! The companions beside him said, O Messenger of God, are we not you brothers? The Prophet replied, You are my companions; my brothers have yet to come.

As in the Hadiths above, here also Prophet Muhammad (PBUH) shows his longing to see his brothers that will come after him, while he was passing beside the graveyard of Medina with some of the companions. For it will not be possible for him to see them in the world. They will come into the stage of life at any time God wills after the Prophet passed away. It will be possible to meet them in the Hereafter.

The companions asked, Are we not your brothers? in order to reduce his longing a bit, as they knew their closeness to him. However, the Prophet called them companion and the others brother. As the ones who saw and talked to him at the time of the Messenger are remembered as companions and the word companion is often used for them; the word brother will often be used among the ones who come after him and they will possibly address one another as brother! no matter what origin or from what country they are. They will love each other more than real brothers through Islam and faith, the essence of their way will be brotherhood, and they will become for one another the most self-sacrificing brother and the most trustworthy friend. In other words, as the most important quality and the most crucial principle of their way is brotherhood of Islam, the Prophet called them brother. There is a sign here that the end of these people will be good and that they will enter the grave with faith. (3)

When the Prophet said: My brothers have yet to come. as they must have thought in the Hereafter, how the Prophet would recognize the people yet to come to the world without having seen them for once, the companions asked:
Then, O Messenger! How will you recognize these people from your community who have yet to come?

Whereupon, the Prophet asked:
If someone has a horse with white signs on the forehead and the feet, can he recognize it among black and gray horses?

The companions replied:
Yes.

That Prophet Muhammad (PBUH) likened these fortunate people to horses with white signs on the forehead and the feet shows that these people are the ones who continually perform their five time-prayers and who abandon big sins. As is stated in the verse (4), on their foreheads there will be a light and whiteness as a mark and sign of their faiths and prostrations.

The hadith continues:
Brothers of mine will come to the Hereafter with shining organs of ablution because of the light that came from their ablutions. And I wait for them beside my Pool of Kawsar. Be aware that some people are driven away from my Pool of Kawsar like bewildered camels. I call out to them: Come here! It is said to me: They changed their attitudes and behaviors after you. Then I say, They are perished; they are perished! (6)

Sources:
1. Mecmaût-Tefasir (Lubabut-Tevil) III, 267; Hak Dini IV, 2731; Tefsirul-Kuranil-Azim, II, 422-423; III, 291 (In the explanation to the 35th verse of An-Nur Surah, the explanation about the person upon light is made: His words are light; his deeds are light; his destination is light; where he leaves is light, in the Hereafter his return will be to light, to the paradise. See also Hak Dini IV, 2730; In relation to the 64th verse of Yunus Surah, believers and friends of God with faces full of light and who are on pulpits of light are mentioned. See also Tefsirul-Kuranil-Azim II, 423. Every pious is the friend of God. Again, it is stated that it is they whose faces are full of light. Also, III, 291: These friends of God, their words are light; their deeds are light; their destinations are light, where they leave is light; and the paradise, where they will go, is also light.
2. Tefsirul Kuranil-Azim II, 422; Hak Dini IV, 2731; Riyazus-Salihin p. 874.
3. As at the end of this hadith, it is mentioned that these people will come to the Pool of Kawsar, it is understood that they are among the fortunate who will enter the grave with faith.
4. Al-Fath Surah, 48; 29.
5. Ankabut Surah, 29;45.
6. el-Tac no. 1645 (Muslim, Nesei); Kırk Hadis, p. 27.