Praise be to Allaah.
This is mentioned in some ahaadeeth, but there is some scholarly difference of opinion as to the soundness of those ahaadeeth. There is no saheeh hadeeth concerning the virtue of the night of the fifteenth of Sha'baan.
It was narrated from Abu Moosa al-Ash'ari that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: Allaah looks down on the night of the fifteenth of Sha'baan and forgives all his creation except a mushrik or one who harbours hatred against the Muslims. Narrated by Ibn Maajah, 1390.
The one who harbours hatred against the Muslims means one who has enmity towards a Muslim brother.
In al-Zawaa'id it says: Its isnaad is da'eef, because Abd-Allaah ibn Lahee'ah is da'eef (weak) and al-Waleed ibn Muslim is mudallis.
There is also some idtiraab (weakness) in the hadeeth, as stated by al-Daaraqutni in al-Ilal, 6/50, 51. He said: This hadeeth is not proven.
It was narrated from Mu'aadh ibn Jabal, Aa'ishah, Abu Hurayrah, Abu Tha'labah al-Khushani and others, but the isnaads are not free of some weakness, and some of them are very weak.
Ibn Rajab al-Hanbali said:
Concerning the virtue of the night of the fifteenth of Sha'baan there are numerous ahaadeeth, concerning which the scholars differed, but most of them classed them as da'eef, and Ibn Hibbaan classed some of them as saheeh.
Lataa'if al-Ma'aazif, 261.
Allaah's descending to the first heaven does not only happen on the night of the fifteenth of Sha'baan, rather it is proven in al-Saheehayn and elsewhere that Allaah descends to the first heaven every night, in the last third of the night. The night of the fifteenth of Sha'baan is included in this general meaning.
Hence, when Abd-Allaah ibn al-Mubaarak was asked about the descent of Allaah on the night of the fifteenth of Sha'baan, he said to the one who asked him: O weak one! The night of the fifteenth?! He descends every night!
Narrated by Abu Uthmaan al-Saabooni in I'tiqaad Ahl al-Sunnah, no. 92.
Al-Aqeeli (may Allaah have mercy on him) said:
With regard to Allaah's descending on the night of the fifteenth of Sha'baan there are ahaadeeth which are weak, but the reports that He descends every night are proven and saheeh, so the night of the fifteenth of Sha'baan is included in that, in sha Allaah.
See also the answer to question no. 8907.
On this site you can also find an article by Shaykh Ibn Baz (may Allaah have mercy on him) concerning the ruling on celebrating the night of the fifteenth of Sha'baan. You can find it under the heading, Periodic Topics (under Useful Material on the Islam Q&A Library page).