The First, The Pre-Existing
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Awwal (in Arabic: ٱلأَوَّلُ), He is The One whose existence is without beginning. He is truly the first, and nothing was before Him.
From the root hamza-waw-lam (أ و ل), which has the following classical Arabic connotations: to be first, principle, to be before, to come back to, to return, to precede, to be the first part, to be foremost.
Tawhid & Tawakkul:
The understanding and belief in the name establishes tawheed (oneness of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ). Al-Awwal means He was first or is the first. But this does not mean that there was nothing before. Rather the beginning was Him. He exists without a beginning and is the source from which everything comes into existence. From this name, we see how He is from ever and forever, and He is never absent and always present. It is a reminder we should rely upon Him alone (the concept of tawakkul).
The First & The Last:
Al-Awwal also implies an opposite. If there is a first, then there must be a last, and this is the name Al-Akhir. This does not mean that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ has a beginning and an end. Instead, it tells us He is the start, and He is the end. It's the race of life; we start with Him and return to Him. In this race, some may try to take shortcuts only to get lost, and the wiser pace themselves, knowing it's a marathon. A few may try to trip or sabotage others to gain an advantage, but they fail to see the race is within themselves. There is no first or second-place prize. At the end of the journey, the satisfied will be those who gave their best effort; they struggled beautifully and gave it their all. For them will be the prize of the akhirah, Insha-Allah. You don't want to have reached the end of the race to realize you had much more to offer; you want to be tired at the end.
In The Name & the Named, Sheikh Tosun Bayrak writes, "It is proper to read, recite, and consider the attributes "The First" and "The Last" together because the meaning is like a circle where the first and the last are one." This interpretation seems to describe these two names accurately. Imam Ghazali mentions something similar, "So He is last with respect to wayfaring [there is nothing that will succeed him], and first with respect to existence: the first beginning was from Him; and to Him is the last return and destination."
Huwal Awwalu wal'Aakhiru waz Zaahiru wal Baatinu wa huwa bikulli shai'in Aleem
"He is the First and the Last, the Ascendant and the Intimate, and He is, of all things, Knowing." — (Qur'an 57:3)
Allazeena izaaa asaabathum museebatun qaalooo innaa lillaahi wa innaaa ilaihi raaji'oon
"Who, when disaster strikes them, say, 'Indeed we belong to Allah, and indeed to Him we will return.'" — (Qur'an 2:156)
The case of the curious mind:
It is narrated on the authority of Abu Huraira that the Messenger of Allah ﷺ said: Men will continue to question one another till this is propounded: Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ created all things but who created Allah? He who found himself confronted with such a situation should say: I affirm my faith in Allah. 
To those with a curious minds, we can't fathom how something can exist from nothing. We're so ingrained by our experiences as to seeing how one thing gives birth to another that we ask the same question of God. But as Prophet ﷺ tells the companions who ask Him this question, he replies that it's a matter of faith. The Qur'an addresses the hypocrites and the disbelievers in many verses. For example:
Awa lam yakfihim annaaa anzalnaa 'alaikal kitaaba yutlaa 'alaihim; inna fee zaalika larahmatanw wa zikraa liqawminy yu'minoon (section 5)
"And is it not sufficient for them that We revealed to you the Book which is recited to them? Indeed in that is a mercy and reminder for a people who believe." — (Qur'an 29:51)
The human mind has many limitations, and we often fool ourselves. We don't know what we don't know, and we can't know the limits of how our intelligence may be bounded. A lot of psychological work has been done to show the many ways in which we're unreliable narrators. One such fallacy is the illusion of explanatory depth (IOED). If we look at the simplest of things we use every day and try to explain in detail how they work, we quickly realize that much of our knowledge is surface level. For example, what's a toilet? We likely answer with confidence what seems to be a silly question. Of course, we know what it is and how to operate one, but when you press the flush button, what's really happening underneath the tank? That's what makes the toilet a toilet. It seems like a strange analogy, but the point is we are often unaware of our ignorance, yet we ask irrelevant questions of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ that we cannot comprehend. And so, the answer Prophet Muhammad ﷺ gives just may be the most intelligent answer to the question.
Supplication from hadith using these two names:
There is a nice du'a mentioned in the hadith literature, which tells us what he would sometimes recite before going to bed. In this du'a, it asks for debts and desires to be cleared so he may rest in peace. He ﷺ calls upon these two names Al-Awwal and Al-Akhir, and we've highlighted them for your reference.
Suhail reported that Abu Salih used to command us (in these words): When any one of you intends to go to sleep, he should lie on the bed on his right side and then say: 
Allāhumma rabba as-samāwāti wa rabbal-arḍi wa rabbal-'arshi al-'aẓīm. Rabbanā wa rabbakulli shay'in, fāliqa al-ḥabbi wan-nawā wa munzila at-tawrāti wal-injīli wal-furqāni, a'ūdhu bika min sharri kulli shay'in anta ākhidhun bināṣiyatihi. Allāhumma anta al-awwalu falaysa qablika shay'un, wa anta al-ākhiru falaysa ba'daka shay'un, wa anta aẓ-ẓāhiru falaysa fawqaka shay'un, wa anta al-bāṭinu falaysa dūnaka shay'un. Iqdhi 'annā ad-dayna wa aghninā min al-faqr.
"O Allah. the Lord of the Heavens and the Lord of the Earth and Lord of the Magnificent Throne, our Lord, and the Lord of evervthing, the Splitter of the grain of corn and the datestone (or fruit kernal), the Revealer of Torah and Injil (Bible) and Criterion (the Holy Qur'an), I seek refuge in Thee from the evil of every - thing Thou art to sieze by the forelock (Thou hast perfect control over it). O Allah, Thou art the First, there is naught before Thee, and Thou art the Last and there is naught after Thee, and Thou art Evident and there is nothing above Thee, and Thou art Innermost and there is nothing beyond Thee. Remove the burden of debt from us and relieve us from want."