The Shakhas of the Vedas are editions of the original (mool) Vedas, propounded by different Rishis. These shakhas are known by the name of these Rishis and preserved through the guru shishya ved paath paddhati.
The mool Veda are four. They are 'apourishiya' (not made by humans). The Rishis compiled the mool Vedas (with or without variations in the mantras) along with the Rishi, Devata, Chand, Swar, Mantra numbering (in different ways), etc respective to each mantra of the Veda. Such compilations are called Shakhas. Shakhas can be classified into two types viz., (a) the compilations without any variations in the mantras of the mool Vedas, and (b) the compilations with some variations in the mantras of the mool Vedas. Maharshi Dayanand has accepted the former as Veda Samhitas and the later as Veda Shakha Samhita. According to him, whilst the mantras of the Ved Samhitas are 'apourishiya', the additions and the variations are Rishi made.
The Shakhas help in understanding the meaning of the Vedas. The Rishi who envisions a true meaning(s) of a given mantra is called the Rishi of that Mantra. Hence, the Mantras marked with a given Rishi can be read together to understand the subject(s) of the Rishi. Devata of a given mantra denotes the key subject of that mantra according to the Rishi of the Mantra. Hence the Mantras marked with a given Devata can be read together to understand a given subject(s). Chand and Swar are checks on the purity of the Mantras.
Apparently, the Shakhas has always been prevalent parallel to the mool Veda. However, Ved Vyaas re-organised / re-established the various shakhas of the Vedas before the advent of Kaliyug in order to preserve the Vedas.
According to MahaBhasya of Patanjali, there were 1131 Shakhas (21+101+1000+9) prevalent during his times. “एकविंशतिधा बाह्वृच्यम्। एकशतमध्वर्युशाखाः। सहस्रवर्त्मा सामवेदः। नवधाथर्वणो वेदः।“ पातन्जल महाभाष्य, पस्प-शाह्निक। This includes the 4 Ved Samhitas. However, now only 16 of these Shakhas are available, which are as follows:
1. 5 of ऋग्वेदः शाकल: (Shaakal) (with and without Baal Khilya Sukta), बाष्कल: (Bhaashkal), आश्वलायन: (Aashvalaayana), शांखायन: (Shaankhaayana) and माण्डूकाय (Mandukaaya)।
2. 6 of यजुर्वेदः
a. 3 of शुक्ल-यजुर्वेदः वाजसनेयी (Vaajasaneyi) or माध्यन्दिनी (Maadhyandini) and काण्व (Kaanva)।
b. 3 of कृष्ण-यजुर्वेदः तैत्तिरीय: (Taittiriya), मैत्रायणी (Maitraayani), कठ: (Katha), and कपिष्ठल (Kapishhthala)।
3. 3 of सामवेदः कौथुमीय (Kauthumiya), राणायनीय (Ranaayaniya) and जैमिनीय (Jaiminiya)
4. 2 of अथर्ववेदः शौनक (Shaunak) and पैप्लाद: (Paipalaada)।
Maharshi Dayanand Saraswati has considered above 4 underlined Shakhas as Veda Samhita, iterating as below:
1. ऋग्वेदः शाकल: (Shaakal) (with Baal Khilya Sukta)
2. यजुर्वेदः वाजसनेयी (Vaajasaneyi)
3. सामवेदः कौथुमीय (Kauthumiya)
4. अथर्ववेदः शौनक (Shaunak)