The story of the Al-Miraj goes that the beloved Prophet (PBUH) was asleep one night towards the end of his prophetic course in Makkah when he was wakened by the angel Gabriel(as) who cleansed his heart before bidding him alight on a angelic horse named Buraq.
Sahih Muslim, Vol. 1, p. 101
I was brought al-Burg who is an animal white and long, larger than a donkey but smaller than a mule, who would place his hoof at a distance equal to the range of vision. I mounted it and came to the Temple (Bait-ul Maqdis in Jerusalem), then tethered it to the ring used by the prophets.
The journey from Makkah to Jerusalem is known as al-Isra, "the night journey". At Jerusalem the Prophet(PBUH) was tested in the following way by Gabriel(as).
Sahih al-Bukhari, Vol. 6, p. 196
Allah's Apostle was presented with two cups, one containing wine and the other milk on the night of his night journey at Jerusalem. He looked at it and took the milk. Gabriel said, "Thanks to Allah Who guided you to the Fitra (i.e. Islam); if you had taken the wine, your followers would have gone astray".
After this began al-Mi'raj, "the ascent". The Prophet(PBUH) passed the sea of kawthar, the sea of "abundance", and then met various prophets, from Adam(as) to Abraham(as), as well as a variety of angels as he passed through the seven heavens. After this Gabriel(as) took him to the heavenly lote-tree on the boundary of the heavens before the throne of Allah.
Sahih al-Bukhari, Vol. 5, p. 147
Then I was made to ascend to Sidrat-ul-Muntaha (i.e. the lote-tree of the utmost boundary). Behold! Its fruits were like the jars of Hajr (i.e. a place near Medina) and its leaves were as big as the ears of elephants. Gabriel said, "This is the lote-tree of the utmost boundary".
Gabriel(as) and Buraq could go no further but the Beloved Prophet(PBUH) went on to the presence of Allah where he was commanded to order the Muslims to pray fifty times a day:
Sahih al-Bukhari, Vol. 1, p. 213
Then Allah enjoined fifty prayers on my followers. When I returned with this order of Allah, I passed by Moses who asked me, "What has Allah enjoined on your followers?" I replied, "He has enjoined fifty prayers on them". Moses said "Go back to your Lord (and appeal for reduction) for your followers will not be able to bear it".
The Prophet(PBUH) went back and forth between Allah and Moses(as) till the prayers were reduced to five per day. Moses(as) then told him to seek yet a further reduction but The prophet stopped at this point and answered Moses:
Ibn Ishaq, Sirat Rasulullah, p. 187
I replied that I had been back to my Lord and asked him to reduce the number until I was ashamed, and I would not do it again.
Allah then said whoever observed the five times of prayer daily would receive the reward of fifty prayers. The Prophet(PBUH) then saw some of the delights of paradise as he returned to Gabriel(as) and Buraq and then beheld the torments of the damned before going back to his home in Makkah that same night. This, briefly, is the narrative of the ascent.
The Qur'an also supports this miraculous journey.
Glory to (God) Who did take His Servant for a Journey by night from the Sacred Mosque to the Farthest Mosque whose precincts We did bless, - in order that We might show him some of Our Signs: for He is the One Who heareth and seeth (all things). Surah 17.1
The "Sacred Mosque" (al-masjidul-haram) is interpreted to be the Ka'aba at Makkah and the "Farthest Mosque" (al-masjidul- aqsa) the Temple at Jerusalem. The great mosque which presently stands next to the Dome of the Rock is accordingly known today as the "al-Aqsa" mosque.
