Imam an-Nasa-i or an-Nisa-i in first place compiled his book as-Sunan al-Kubra السنن الكبرى in which he collected sahih and non-sahih narrations (10770 in total) and which can be considered as the source of the ahadith in his other book as-Sunan as-Sughra السنن الصغرى (5774 in total) which is also often referred to as al-Mujtaba (the collected, picked or chosen) المجتبى or al-Mujtana (the reaped or picked -mostly used in context of fruits in Arabic-) المجتنى which is a summary of his as-Sunan al-Kubra and includes what an-Nasa-i considered as sahih (this book is the soundest among the Sunan compilations and the biggest one two) and there's no single fabricated narration inside this book as many scholars say. Imam an-Nasa-i's conditions on narrators are considered as more strict than those of imam Muslim (refer to What exactly are the conditions of al-Bukhari and Muslim?).
It is a discussed matter whether an-Nasa-i himself summarized his as-Sunan as-Sughra or his student ibn as-Sunni.
Imam a-Dahahabi quoted ibn Athir's narration on the compilation or extraction of as-Sunan as-Sughra from as-Sunan al-Kubra:
In the following I'll be translating from Arabic, as this is my own translation take it carefully!
قال ابن الأثير : وسأل أمير أبا عبد الرحمن عن سننه : أصحيح كله ؟
قال : لا . قال : فاكتب لنا منه الصحيح . فجرد المجتنى .
Ibn al-Athir said: An emir asked abu 'Abdullah (referring to an-Nasa-i) about his Sunan: Is it all sahih? and he answered: no. The emir than said: So arrange for us a book the sahih from it. And he extracted al-Mujtaba.
قلت : هذا لم يصح ; بل المجتنى اختيار ابن السني .
I (referring to a-Dhahabi) say: this claim has no backup or is even wrong, but al-Mujtaba was the choice ibn as-Sunni. (Source: Siyar a'laam an-Nubala' سير أعلام النبلاء)
The scholar ibn Amin (see here) contradicts him by saying that the version of ibn as-Sunni is only one of the narrations of the book, as some old copies of the book seem to indicate (the authors of copies of the 6th Hijri century gave a chain as a reference in which is quoted that ibn as-Sunni ابن السني narrated from an-Nasa-i and also added the remark that the book was compiled and arranged by an-Nasa-i). Here a copy of the quote
تأليف أبي عبد الرحمن أحمد بن شعيب بن بحر النسائي. رواية أبي بكر أحمد بن إسحاق بن السني عنه. رواية القاضي أبي نصر أحمد بن الحسن بن الكسار عنه. رواية الشيخ أبي محمد عبد الرحمن بن محمد الدوني عنه. رواية أبي الحسن سعد الخير بن محمد بن سهل الأنصاري عنه. رواية الشيخ الإمام زين الدين أبي الحسن علي بن إبراهيم بن نجاد الحنبلي الواعظ.
And this is the book often referred to as-Sunan an-Nasa-I and which was translated on for example sunnah.com and commented by many scholars such as as-Suyuti (called زهر الرُّبى على المجتبى ) and as-Sindi.
See also the fatwa's on islamqa #171124 (so far only available in Arabic) and #171895 on imam an-Nasa-i or an-Nisa-i and as-Sunan as-Sughra.