What is Alif Laam Mim

Twenty-nine chapters of the Quran begin with “al-huroofe-Muqatiaat” (letter symbols). Following is a list of these chapters with the symbols, in the order they appear in the Quran: 

Chapter 2, The Cow: Alif Laam Mim

Chapter 3, Ale-Imran: Alif Laam Mim

Chapter 7, al-A’araaf: Alif Laam Mim Suad

Chapter 10, Yunus: Alif Laam Ra

Chapter 11, Hud: Alif Laam Ra

Chapter 12, Yusuf: Alif Laam Ra

Chapter 13, Ra’d: Alif Laam Mim Ra

Chapter 14, Ibrahim: Alif Laam Ra

Chapter 15, Hijr: Alif Laam Ra

Chapter 19, Maryam: Kaf Ha Ya Ain Suad

Chapter 20, Ta-Ha: Ta Ha

Chapter 26, The Poets: Ta Sin Mim

Chapter 27, The Ant: Ta Sin

Chapter 28, Qasas: Ta Sin Mim

Chapter 29, The Spider: Alif Laam Mim

Chapter 30, The Romans: Alif Laam Mim

Chapter 31, Luqman: Alif Laam Mim

Chapter 32, The Adoration: Alif Laam Mim

Chapter 36, Ya-Sin: Ya Sin

Chapter 38, Suad: Suad

Chapter 40, The Believer: Ha Mim

Chapter 41, Ha-Mim: Ha Mim

Chapter 42, Shuraa: Ha Mim; Ain Sin Qaf

Chapter 43, The Embelishment: Ha Mim

Chapter 44, The Smoke: Ha Mim

Chapter 45, The Kneeling: Ha Mim

Chapter 46, The Sandhills: Ha Mim

Chapter 50, Qaf: Qaf

Chapter 68, The Pen: Nun  

From the list one can notice that eight chapters begin with the Arabic letters Alif Laam Mim. Some scholars believe, 

“…those suras (chapters) of Holy Qur’an that start with huroofe-muqatiaat ( letter symbols) have a common link between their context through these letters. In other words, those letter symbols that are common in different chapters tell us that the context of those surahs has some common link also. The proof of the above statement is the link and the similarity that can be seen between this surah (Maryam) and surah Suad in which it talks about the story of the Prophets.” The same is the case with the chapter 26 (The Poets), chapter 27 (The Ant), chapter 28 (al-Qasas) and chapter 20 (Ta-Ha)  in the story of Moses (a.s.) and his encounter with the Pharaoh and the link is through the letter “Ta” which is the common letter in all these four chapters.” (Source)

Some have explained that each letter refers to a positive quality and so for example, Alif Laam and Mim refer to 

Alif is for Allah  

Laam is for Latif (name of Allah)  

Mim is for Malik (another name of Allah)  

Modern researchers have interesting points of view as well.  

Amin Ahsan Islahi, a renowned exegete of the Quran, has mentioned that Arabs used to use such letters in their poetry and since Quran addressed them in their own linguistic style, it was only appropriate for Quran to use the same style. He agrees with Razi and mentions that since these letters are names for Surahs, being proper nouns they are not bound to have a meaning. At the same time, he cites research from Hamiduddin Farahi, a Quranic scholar from the Indian subcontinent, on how these letters must be appropriately chosen according to the content and theme of the surahs. Farahi links these letters back to Hebrew alphabet and suggests that those letters not only represented phonetic sounds but also contained a symbolic meaning to them, and Quran perhaps uses the same meanings when choosing the letters for surahs. For instance, in support of his opinion, he presents the letter Nun (ن), which symbolizes fish and Surah Nun mentions Prophet Jonah as ‘companion of the fish’. Similarly, the letter Tuay (ط) represents a serpent and all the Surahs that begin with this letter mention the story of Prophet Moses and serpents.  

Western scholars have only occasionally attempted to explain them. In 1996, Keith Massey proposed new evidence for an older theory that the “Mystery Letters” were the initials or monograms of the scribes that had originally written the sura down. As evidence for this, he demonstrated that the letters themselves occur in a specific order, suggesting a hierarchy of importance. This idea has not yet gained wide acceptance. Other explanations similarly failed to satisfactorily explain the letters. (Source) 

For more information read this and this.