Towards the end of the SandhyAvandanA, I remember reciting the following:
bhArgava, chyavana, ApnavAn, aurava, jamadajnya, pancha rShi pravarAnvita
AshvalAyana sUtra, shAkala shAKhAdhAyih
ahaM bho abhivAdaye
Which translates to:
Hailing from the lineage of the five excellent rShi’s namely BhArgava, chyavana, ApnavAn, aurava, jamadajnya,
Born into the vatsa GOtra,
following AshvalAyana’s sutras, student of the the shAkala shAKha,
I Gauthama Sharma,
Offer my Salutations.
This abhivAdane is used by a Brahmin while introducing himself to another person. It provides details such as the Pravara and Gotra i.e information about the lineage, the name of the SutrakAra (Author) whose laws he has been following, the Veda Shakha that he has studied/is studying.
Of these, I knew the meaning of Gotras and Pravaras at the time of my UpanayanA. However, I had no idea about the the Sutra, nor about the Veda Shakha. The former, I am yet to understand completely, but regarding the latter, I have managed to gather some information in these past couple of months.
During the dvApara yuga, a sage named Krishna DvaipAyana, who was well versed with the Vedas realized that as the commencement of the new Yuga (Kali Yuga), the mental faculties of men would decline. Hence an individual Brahmin would not be able to remember and master all the Veda mantras. But Veda mantras were the very basis for humans to realize the eternal truth. Veda mantras contained in them not only the rules for outwardly conduct but they also served as guide to inner purification. Hence to aid mankind, this kind sage classified the Vedas into four divisions. They are:
- Rg Veda whose mantras are conducive to worship or prayer,
- Yajur Veda whose mantras portray the ritualistic and yajna procedures,
- Shukla Yajur Veda
- Krishna Yajur Veda
- Sama Veda, which are hymns in musical form
- Atharva Veda, which stress the performance of yajnas and contain mantras designed to protect men from dangers and enemies.
Owing to this super-human classification effort, sage Krishna DvaipAyana obtained the name Veda Vyasa. Vaishnavas believe that Lord Vishnu assumed the AvatAr of Veda Vyasa for the purpose of organizing human knowledge for the benefit of mankind.
Lord Veda Vyasa further divided each of the Vedas into several sections or branches. Each branch is known as a shAkha. Thus a Brahmin of the latter age was expected to know one Shakha of any of the Vedas at the least. There was no limitation on how many shAkhas a person could master. There were many who did manage to master more than one Shakhas. Some of the common North Indian surnames indicate the proficiency of a person in two or more Vedas. Dvivedi, Dubey, Dave were the ones who mastered two of the Vedas. Trivedis and Tiwaris mastered three of the Vedas. Chaturvedis , Chattopadhyayas were well versed with all the four Vedas. 
Each Veda shAkha consists of the following parts:
- A Samhita
- A Brahmana
- An Aranyaka
- An Upanishad.
When Veda VyAsa made this classification, there were 1180 Veda Shakhas in total . Of these 1180,
- 21 belonged to the Rg Veda
- 109 belonged to the Yajur Veda
- 15 from Shukla Yajur Veda
- 94 from Krishna Yajur Veda
- 1000 belonged to Sama Veda
- 50 were from the Atharvana Veda.
Unfortunately in the present day, we have lost a lot many of these Shakhas due to lack of practice.
Among the 21 Rg Veda Shakhas, apparently only one is in active practice today. And that is the shAkala shAKha, to which my family belongs. In fact most of the Gaud Saraswat Brahmins belong to this shAkha. The Brahmana, Aranyaka and the Upanishad of this Shakha all go by the name Aitareya. Hence the Shakha is also referred to as Aitareya Shakha. However, there is still a claim that the sAnkhyAyana shAkhA is still known to a few vEdapathis in Uttar Pradesh and Gujarat, but this is not certain. sAnkhyAyana Shakha has Kaushitaki BrAhmana, Kaushitaki AranyakA and KaushItaki Upanishad
Of the 15 Shakhas of the Shukla Yajur Veda, only two are known. Them being kAnva shAkha and the mAdyAndina shAkha.
Of the 94 Shakhas of the Krishna Yajur Veda, the Taitreeya Shakha is fully available and is in active use in South India, while another shAkha named the maitrAyanIya Shakha manages to survive in Maharastra.
Of the 1000 Shakhas of the SAma VEdA, we have lost 997 Shakhas. The Surviving shAkhas are talavakAra ShAkha of the Jaimini school, the rAnAyanIya shAkha and the Kouthuma Shakha.
At one point in time, it was feared that all the 50 Shakhas of Yajurveda had gone extinct. However, one shAkha was found to be in practice and continues to survive today. It goes by the name Sownaka ShAkha. Most of the samhitAs of the Atharva VEda are lost. The only BrAhmana of this Veda that’s available today is the gopatha BrAhmana. Among the Upanishads of this shAkha, Prasna, Mundaka, mAndUkya as well as the Nrisimha thApini Upanishad are available and are actively studied 
In the present times, the sUtras and ShAkhas seem to have lost their importance. Most people don’t even know about their sUtras and shAkhas, let alone knowing their contents. Unfortunately, I belong to this category of people.
Looking at the the interest (or the lack thereof) which the current generation of Brahmins is showing in the study of their scriptures, these remaining few vEda shAkhAs run the risk of becoming extinct. One of the main duties of the various Mutts is to work towards preservation of this ancient knowledge. However, it’s interesting to note that some people simply categorize them as Hindutva Mutts and easily get away with it.
Sigh! Sorry state of affairs indeed.
 Sri Chandrasekharendra Saraswati, The Vedas, Bhavan’s Book University, 7ed, pg 110
 Ibid, pg 112
 Ibid, pg 113
 Ibid, pg 111
 Kireet Joshi, The Veda And Indian Culture:an Introductory Essay, Motilal Banarasidass Pub, ISBN:8120808894, Pg 91