A Sith apprentice was an individual (and on occasion, a child), that began serious training under a Sith Master, usually chosen by the Master himself or herself, after the potential apprentice displayed an impressive act of loyalty or cruelty. In some cases, the individual pledged themselves to the teachings of the Sith Master, thereby becoming the apprentice. In other situations, the Master sought out the student, being guided by the will of the dark side of the Force. Sith apprentices were analogous to JediPadawans, and the relationship between Sith Master and apprentice was the analogue to Jedi Knight and Padawan. A Sith apprentice's time as a student ended either with his or her death, or when the apprentice killed the master, taking on an apprentice of his or her own, thereby ascending to the rank of Sith Master.
The role of a Sith apprentice was different from that of a Jediapprentice. While a Padawan often accompanied their Master on whatever missions they partook, Sith Masters often sent their apprentices on missions alone. The Sith apprentice acted as an agent of the Master, doing their bidding and carrying out their will without question. It was not uncommon for the apprentice to face death, and even succumb to it while carrying out their Master's orders. In addition, a Jedi Padawan was taught by the master, whereas one studying under a Sith Master was to learn all they could from them; teaching was not necessarily the Master's responsibility.
While Sith apprentices were always under the direction of their Masters, oftentimes they were allowed to pursue or birth their own schemes and machinations, further preparing them to successfully assume the rank of Sith Master themselves.
Sith apprentice training was both similar and very different to Jedi training. Sith apprentices learned from their Master's lightsabertechnique, and the ways of the Force: specifically the dark side and how to harness it. Sith apprentices were taught to staunchly adhere to the Sith code.
"Of all the monsters I have created, I still regard Darth Vader as something of a minor masterpiece. No, he was not an entirely alchemical creation, but he was my monster nevertheless. Even though he failed to live up to his full potential, there was much pleasure in transforming Anakin Skywalker from a bright-eyed, tousel-headed youth into the greatest Jedi killer of all time. Yes, he ultimately turned against his Master, as monsters sometimes do, but that was my fault, not his. Given the opportunity to create Vader again, I would, and with zeal."
In Darth Bane's New Sith Order and the Rule of Two, the Sith apprentice and the Sith Master both held the title Dark Lord of the Sith simultaneously, unlike the earlier tradition of a sole Dark Lord (though it appeared that the apprentice did not assume the role of Dark Lord until their training was complete).