I have heard there is a hadith about 73 sects from Ibn Majah. I want to know the authenticity of this hadeeth. Please give me some reasons to prove this hadeeth as sahih.
The hadith you are referring to is authentic, and has multiple versions with different grades.
It was compiled in Jami' at-Tirmidhi hadiths 2640 and 2641, Sunan Abi Dawud hadiths 4596 and 4597, Sunan Ibn Majah hadiths 3991 and 3992 and 3993, Musnad Ahmad hadith 2/332, Al-Mustadrak hadith 61, as well as numerous other books of hadith.
The hadith was narrated with essentially two versions through the authority of multiple companions:
- 'Abdullah ibn 'Amr ibn al-'As
- 'Abdullah ibn Mas'ud
- Abu Hurairah
- Abu Musa al-Ash'ari
- Abu Umamah al Bahili
- Abu al-Darda'
- 'Ali ibn Abi Talib
- 'Amr ibn Awf
- Anas ibn Malik
- 'Awf ibn Malik
- Jaber bin 'Abdullah
- Mu'awiyah ibn Abu Sufyan
- Sa'd ibn Abi Waqqas
- Wathela ibn al-Asqa'
The first version mentions that the Muslim nation will be divided into 73 sects and mention that only one sect will be saved:
عَنْ أَنَسِ بْنِ مَالِكٍ، قَالَ قَالَ رَسُولُ اللَّهِ ـ صلى الله عليه وسلم ـ إِنَّ بَنِي إِسْرَائِيلَ افْتَرَقَتْ عَلَى إِحْدَى وَسَبْعِينَ فِرْقَةً وَإِنَّ أُمَّتِي سَتَفْتَرِقُ عَلَى ثِنْتَيْنِ وَسَبْعِينَ فِرْقَةً كُلُّهَا فِي النَّارِ إِلاَّ وَاحِدَةً وَهِيَ الْجَمَاعَةُ
It was narrated from Anas bin Malik that the Messenger of Allah (ﷺ) said: The Children of Israel split into seventy-one sects, and my nation will split into seventy-two, all of which will be in Hell apart from one, which is the main congregation.
The second version mentions that the Muslim nation will be divided into 73 sects only, with no mention of the sect that will be saved:
عَنْ أَبِي هُرَيْرَةَ قَالَ: قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم: افْتَرَقَتِ الْيَهُودُ عَلَى إِحْدَى أَوْ ثِنْتَيْنِ وَسَبْعِينَ فِرْقَةً وَتَفَرَّقَتِ النَّصَارَى عَلَى إِحْدَى أَوْ ثِنْتَيْنِ وَسَبْعِينَ فِرْقَةً وَتَفْتَرِقُ أُمَّتِي عَلَى ثَلاَثٍ وَسَبْعِينَ فِرْقَةً
Narrated AbuHurayrah: The Prophet (ﷺ) said: The Jews were split up into seventy-one or seventy-two sects, and the Christians were split up into seventy-one or seventy-two sects, and my community will be split up into seventy-three sects.
Note that sect is referring to groups that will differ in their creed ('Aqīda, Arabic: العقيدة). Groups that differ on branches of practice are called jurisprudence schools (madhhab, Arabic: مذهب).
For more information, you may refer to this discussion (Arabic only). The reason the hadith is authentic through some of its chains is that the entire chain is trustworthy, of good memory, and continuous.
The truth is that these narrations are all likely weak, corrupted, or fabricated, and there is no authentic evidence whatsoever for the part that says “all of them will enter the Hellfire except one”.
Famous contemporary hadith scholar, Mohammad Al-Hasan Al-Dido has also said that the part that says “all of them will enter the Hellfire except one” is inauthentic and there is not a single authentic narration containing it. 
The Kuwaiti Islamic scholar Dr. Ḥākim al-Muṭayrī conducted an extensive, 39 page, study of all of the relevant hadith narrations regarding this issue. 
He concluded in his study that:
وعلى كل فكل طرق هذا الحديث مناكير وغرائب ضعيفة ومنكرة، وأحسنها حالا حديث أبي هريرة وهو حديث حسن، مع تساهل كبير في تحسينه لتفرد محمد بن عمرو به، وهو صدوق له أوهام خاصة في روايته عن أبي سلمة عن أبي هريرة، ولهذا كان القدماء يتقون حديثه كما قال يحيى بن معين.
All of the ṭuruq (the chains of narrators) of this hadith are objectionable and unauthentic. The best of them is the hadith of Abū Hurayra, which is a ḥasan hadith (i.e. not good enough to be considered authentic, but having an acceptable meaning and not clearly fabricated), provided that we extend it great latitude (i.e. lower our standards) for the fact of Muhammad bin ʿAmr being its only transmitter, who is known to be a truthful person who has awhām (plural of wahm, "confusion" or "delusion", meaning he gets confused and mixes up narrations), especially in his narrations from Abū Salama, from Abū Hurayra, and for this reason the early hadith scholars were cautious of his narrations, as Yaḥyā bin Maʿīn has mentioned.
Please note that the strongest hadith that Dr. al-Muṭayrī mentioned in his conclusion isn't authentic and is a version of the hadith that does NOT have the part that says "all of them will enter the Hellfire except one".
For those who disagree, the study has addressed all the hadiths that you might believe is evidence for this hadith to be authentic and I encourage everyone to check it out.
The hadith is not Sahih and shouldn't be taken seriously within the deen.
Many Muslims have used this inauthentic hadith to commit blatant and most-often unjustified takfir upon other Muslims and cause unnecessary division within the ummah.
Just do your best to follow the deen and may Allah accept from all of us, Ameen.