The Self-Subsisting, The Self-Existing.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Qayyum (in Arabic: ٱلْقَيُّومُ), His essence/nature is to exist being utterly free from dependence on anything or anyone. Al-Qayyum is the one through whom all things arise. He is Ever-Living and Self-Sustaining. Everything else in existence depends on Al-Qayyum.
From the root qaf-waw-mim (ق و م), which has the following classical Arabic connotations: to stand up, to rise up, to ascend, become high, to begin, to start, to come, to pass, to arise, to set out, to be on-going, to exist, to be founded, to be built on, to undertake, to carry out, to manage, to make aright, straighten out, accurate, exact, to make righteous, upright.
The pairing Al-Hayy and Al-Qayyum:
The name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ Al-Qayyum, He is the one who is self-existing. Three verses in the Qur'an include the name Al-Qayyum, and it always occurs in pairs with the name Al-Hayy (the living). Together these names show the attribute of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ being independent and sovereign. He is ever-living not because some source enables Him but because of His essence of self-sustaining / self-subsisting. In contrast, all life remains dependent on one thing or another. They require proper living conditions, which include specific temperatures, the correct atmospheric pressure, composition, and dependence on many other elements. There is no one but Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ that can exist without truly needing anything from anyone. All things in existence rely upon Him for their being.
Allaahu laaa ilaaha illaa Huwal Haiyul Qaiyoom
"Allah – there is no deity except Him, the Ever-Living, the Sustainer of existence." — (Qur'an 3:2)
Wa 'anatil wujoohu lil Haiiyil Qaiyoomi wa qad khaaba man hamala zulmaa
"And [all] faces will be humbled before the Ever-Living, the Sustainer of existence. And he will have failed who carries injustice." — (Qur'an 20:111)
Allahu laaa ilaaha illaa Huwal Haiyul Qaiyoom; laa taakhuzuhoo sinatunw wa laa nawm; lahoo maa fissamaawaati wa maa fil ard; man zal lazee yashfa'u indahooo illaa bi-iznih; ya'lamu maa baina aydeehim wa maa khalfahum wa laa yuheetoona bishai'im min 'ilmihee illaa bimaa shaaa'; wasi'a Kursiyyuhus samaawaati wal arda wa laa Ya'ooduhoo hifzuhumaa; wa Huwal Aliyyul 'Azeem
"Allah – there is no deity except Him, the Ever-Living, the Sustainer of [all] existence. Neither drowsiness overtakes Him nor sleep. To Him belongs whatever is in the heavens and whatever is on the earth. Who is it that can intercede with Him except by His permission? He knows what is [presently] before them and what will be after them, and they encompass not a thing of His knowledge except for what He wills. His Kursi extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High, the Most Great." — (Qur'an 2:255)
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ - The Provider:
There is another understood meaning of Al-Qayyum: He is the one who holds upright or causes to stand up. This connotates a sense of dependence, He not only is self-sustaining but is the sustainer. For example, qa'im (قايم) or qaimatun (قَائِمَةٌ) is an Arabic name/word for standing, rising, or being upright. Al-Qayyum is the more emphatic derivative containing the same root q-w-m. And in Surah Ar-Ra'd ayah 33 it mentions, Afaman Huwa qaaa'imun 'alaa kulli nafsim bimaa kasabat meaning, "Is then He (Allah) Who takes charge (guards, maintains, provides, etc.) of every person and knows all that he has earned (like any other deities who know nothing)?" Whether we realize it or not, we're all being upheld by the willingness and command of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. He is the maintainer who 'guards, protects, and provides.
A man's duty:
The word Qayoom also shares the same root of the word qawama, a concept of survival and subsistence. This topic has become controversial because it plays into gender politics. The verse which is viewed with scrutiny comes from Surah Nisa ayah 34, Arrijaalu qawwaamoona alan nisaaa'i bimaa fad dalallaahu ba'dahum 'alaa ba'dinw wa bimaaa anfaqoo min amwaalihim "Men are in charge of women, because Allah hath made the one of them to excel the other, and because they spend of their property (for the support of women)."
Some translations of this verse use "in charge," which seems more possessive but really it means a duty upon man to provide, protect, and look after the women in their lives. To be a guardian or a supporter. This includes, but is not limited to, material sustenance. But this sustenance can be further extended to emotional support, guidance, and companionship required for a healthy marriage. It's not to say women aren't capable, in studying seerah we know Khadija bint Khuwaylid (ra), the Prophet's ﷺ first wife, was a very successful merchant on her own. Ibn Arabi says to assume this character trait, "If one of the servants meets this condition in responding to the needs of the one who seeks his support, and if it comes frequently, he is self-existing."
Anas bin Malik said: "Whenever a matter would distress him, the Prophet (ﷺ) would say: 'O Living, O Self-Sustaining Sustainer! In Your Mercy do I seek relief (Ya hayyu ya Qayyum, bi-rahmatika astaghith).'” And with this chain, that he said: "The Messenger of Allah (ﷺ) said: 'Be constant with: "O Possessor of Majesty and Honor (Ya Dhal-Jalali wal-Ikram).'" 
The believer benefits from knowing that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Hayy and Al-Qayyum. We acknowledge that reliance upon anything other than Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is futile. No other is ever-living like Al-Hayy, and none is self-subsisting except Al-Qayyum. And so, as Imam Ghazali puts it, "man's access to this attribute is in proportion to his detachment from everything that is not God the most high."
Anas said, "I was with the Prophet, may Allah bless him and grant him peace, when a man made supplication, saying, 'O Originator of the heavens! O Living, Self-Subsistent! I ask of You.' He said, 'Do you know by what he has made supplication? By the One in whose hand my soul is, he has asked Allah by the Name that He answers when He is asked by it." 
 Hasan (Darussalam) Jami at-Tirmidhi 3524
 Sahih (Al-Albani) Al-Adab Al-Mufrad 705