ALLAHU AKBAR (الله اكبر)
Allahu Akbar means 'Allah سبحانه و تعالى is the Greatest', meaning that nothing can supersede Allah سبحانه و تعالى or be better than Him. It is a common phrase in Islam and it features in the adhan(Azan) and prayer. We Muslims tend to use it for lots of reasons and to express many emotions. 
And say, "Praise to Allah سبحانه و تعالى , who has not taken a son and has had no partner in [His] dominion and has no [need of a] protector out of weakness; and glorify Him with [great] glorification." Quran.Surah Al-Isra 17:111.
Whenever we went up a place we would say, "Allahu--Akbar (i.e. Allah سبحانه و تعالى is Greater)", and whenever we went down a place we would say, "Subhan Allah." Sahih Al-Bukhari 2993 (Vol 4:236)
Abu Hurairah رضي الله عنه reported: A man said: "O Messenger of Allah ﷺ, I intend to set out on a journey, so counsel me.'' Muhammad ﷺ said, "Fear Allah سبحانه و تعالى, and ascending towards the raised ground, you should recite: "Allahu Akbar (Allah سبحانه و تعالى is Greatest).'' When the man turned away, he ﷺ said, "O Allah سبحانه و تعالى! Shorten the distance for him and make the journey easy for him.'' At-Tirmidhi 3445.
'Allahu Akbar, Allahu Akbar, Allahu Akbar, Allahu Akbar, Ash hadu an la ilaha illal lah, ash hadu an la ilaha illal lah,
Ash hadu an-na Muhammadar resulul-lah, Ash hadu an-na Muhammadar rasulul-lah, hayya 'alas, salah, hayya 'alas salah,
hayya 'alal falah, hayya 'alal-falah Allahu Akbar; Allahu Akbar, La ilaha illal lah.'
The Prophet ﷺslaughtered two rams as sacrifice and mentioned Allah's سبحانه و تعالى Name and said, "Allahu-Akbar" while slaughtering). Sahih Al-Bukhari 7399 (Vol. 9:496)
"The Messenger of Allah ﷺsaid: 'The key to prayer is purification, its opening is to say 'Allahu Akbar' and its closing is to say As-salamu 'alaikum.'" Sunan Ibn Majah Vol. 1:276.
“We went in the morning on this day with the Messenger of Allah ﷺ from Mina to ‘Arafat. Some of us recited the Takbir (Allahu Akbar) and some of us recited the Tahlil (La ilaha illallah), and neither criticized the other.” Sunan Ibn Majah Vol. 1:3008.
When you go to bed, recite "Subhan Allah' thirty three times, 'Al hamduli l-lah' thirty three times, and 'Allahu Akbar' thirty four times. Sahih al-Bukhari 5362 (Vol. 7:275) 
The dhikr after salaah referred to in the question has been reported in a hadeeth reported by Abu Hurayrah from the Messenger of Allaah ﷺ : “Whoever glorifies Allaah (says: Subhaan Allaah) thirty-three times immediately after each prayer, and praises Allaah (says: Al-hamdu Lillaah) thirty-three times, and magnifies Allaah (says: Allaahu akbar) thirty-three times, this makes ninety-nine, then to complete one hundred says: Laa ilaaha ill-Allaah wahdahu laa shareeka lahu, lahu’l-mulk wa lahu’l-hamd wa huwa ‘ala kulli shay’in qadeer (There is no god except Allaah Alone, with no partner, His is the power and His is the praise, and He is Able to do all things) – his sins will be forgiven even if they are like the foam of the sea.” Sahih Muslim, 537