What Is the Meaning of “La hawla wa la quwwata illa Billah”?
Question

What is the meaning of La hawla wa la quwwata illa Billah? When should we say La hawla wa la quwwata illa Billah?
Praise be to Allah.The meaning of La hawla wa la quwwata illa Billah
The meaning of this phrase, La hawla wa la quwwata illa Billah, there is no power and no strength except with Allah, is a person’s admission that he is unable to do anything without the help and support of Allah. No matter how great his power, energy and strength may become, a person still cannot do anything without the help of Allaah Who is Above all of His creation, the Most Great, compared to Whom nothing else is great. Every strong person is weak when compared with Allah. Every great person is insignificant and weak when compared with His Might.
When should we say “La hawla wa la quwwata illa Billah”? 
This sentence is uttered when some serious matter befalls a person that he cannot cope with, or there is something that is very difficult for him to deal with. (Shaykh Sa’d al-Humayd)
Among the occasions on which this phrase is uttered are the following: 

When turning over at night: 

‘Ubadah ibn al-Samit said: “The Messenger of Allah (peace and blessings of Allah be upon him) said: ‘Whoever turns over at night and says La ilaha ill-Allah wahdahu la shareeka lah, lahu’l-mulk wa lahu’l-hamd wa huwa ‘ala kulli shay’in qadeer. Al-hamdu-lillah wa subhaan-Allah wa la ilaha ill-Allah wa Allahu akbar, wa La hawla wa la quwwata illa Billah (There is no god but Allah Alone, with no partner or associate. To Him belongs praise and His is the sovereignty, and He is able to do all things. Praise be to Allah, glory be to Allah, there is no god but Allah and Allah is Most Great. There is no power and no strength except with Allah), then he says, Allahumma ighfir li (O Allah, forgive me), or he makes du’a, his prayer will be granted. If he does wudu and prays, his salah will be accepted.” (Narrated by al-Bukhari, 1086) 

When the muadhdhin says “Hayya ‘ala’s-salaah (come to prayer)” or “Hayya ‘ala’l-falah (come to success)” 

Hafs ibn ‘Asim ibn ‘Umar ibn al-Khattab narrated from his father that his grandfather ‘Umar ibn al-Khattab said: “The Messenger of Allah (peace and blessings of Allah be upon him) said: ‘When the muadhdhinin says “Allahu akbar, Allahu akbar (Allah is Most Great),” and one of you says, “Allahu akbar, Allahu akbar”; then he says, “Ashhadu an la ilaha illa Allah (I bear witness that there is no god except Allah” and you say “Ashhadu an la ilaha illa Allah”; then he says, “Ashhadu anna Muhammadan Rasool Allah (I bear witness that Muhammad is the Messenger of Allah” and you say, “Ashhadu anna Muhammadan Rasool Allah”; then he says, “Hayya ‘ala’s-salah (come to prayer)” and you say, “La hawla wa la quwwata illa Billah”; then he says, “Hayya ‘ala’l-falah (come to success)” and you say, “La hawla wa la quwwata illa Billah”; then he says “Allahu akbar, Allahu akbar,” and you say, “Allahu akbar, Allahu akbar”; then he says “La ilaha illa Allah” and you say, “La ilaha illa Allah” – if he says that from the heart, he will enter Paradise.’” (Narrated by Muslim in his Saheeh, 578, and by Abu Dawood in his Sunan, 443)

When leaving one's house 

Anas ibn Malik said: “The Messenger of Allah (peace and blessings of Allah be upon him) said: ‘Whoever says – i.e., when he is leaving his house – “Bismillah, tawwakaltu ‘ala Allah, La hawla wa la quwwata illa Billah (In the name of Allah, I put my trust in Allah, and there is no power and no strength except with Allah),” it will be said to him, “You are taken care of and protected,” and the Shaytan will leave him alone.’” (Narrated by al-Tirmidhi in his Sunan, 3348. Abu ‘Isa said, this is a hasan saheeh ghareeb hadeeth which we only know through this isnad. See Saheeh al-Jami’ by al-Albani, 6419. It was also narrated by Abu Dawood in his Sunan (4431), where he adds: “And another shaytan says to him, ‘What can you do with a man who has been guided, taken care of and protected?’”) 

After praying 

Abu’l-Zubayr said: “Ibn al-Zubayr used to say immediately after every prayer, when he had said the salaam, ‘La ilaha ill-Allah wahdahu laa shareeka lah, lahu’l-mulk wa lahu’l-hamd wa huwa ‘ala kulli shay’in qadeer. La hawla wa la quwwata illa Billah wa la na’budu illa iyyah. Lahu’l-ni’mah wa lahu’l-fadl wa lahu’l-thana’ al-hasan. La ilaha ill-Allah mukhliseena lahu’l-deena wa law kariha’l-kaafiroon (There is no god but Allah Alone, with no partner or associate His is the sovereignty and to Him be praise, and He is able to do all things. There is no power and no strength except with Allah, and we worship none but Him.From Him (alone) come all blessings and favours, and all good praise is due to Him. There is no god but Allah and we make our worship purely for Him (alone) however much the disbelievers may hate that).’ He said: ‘The Messenger of Allah (peace and blessings of Allah be upon him) used to say these words immediately after every prayer.’” (Narrated by Muslim in his Saheeh, 935)
For other issues related to adhkar, please see these answers: 112098 , 105396 , and 147123 .
And Allah knows best.