“Taraweeh” by definition means something that involves a rest and, interestingly, Taraweeh has become very arduous among Muslims. Some Muslims will insist it has to be a certain number of rakats and so on, but there's a lot of flexibility with this. It is mentioned in a narrative that the Prophet, peace be upon him, came out and prayed with his community on one of the nights of Ramadan and then people told each other. The next night, more people came and then, even more people came. Eventually, the Prophet, peace be upon him, did not come out to pray with the community. They were making signals from the mosque to get him to come out, at least to let him know that they were there and waiting for him to come and pray with them. However, the Prophet, peace be upon him, left them to pray on their own. In the morning when he came out for the regular morning prayers, it is related that he said to them, "I knew of your presence, but I deliberately refrained from coming out because I didn't want this to become obligatory on you."
The explanation of this given by scholars is that if the Prophet, peace be upon him, did this regularly with them and became comfortable with them, that practice might be seen as an obligatory practice among Muslims.
If you think of that being passed on generation after generation, later on, people might not be able to do that. It's better that the Prophet, peace be upon him, left it as kind of optional practice. The narratives explain that it remained like that in the time of Abu Bakr who was the first caliph and then during the time of Umar, who was the next caliph in line. It is reported that Muslims were praying in separate groups in the mosque. Umar the caliph saw it apt to gather everybody together and let them be led by one reciter.
It's not clear when it started to be called Taraweeh prayers, but this is when most scholars will say it became a formal and regular practice among Muslims. To pray at least behind one reciter, as opposed to doing it informally in various groups: a group coming, praying, leaving, another group coming, praying, and leaving. Maybe even two different groups praying in the mosque at one time behind separate leaders. However, it became a unified practice in the time of Umar and from thence onwards, it was established as a regular and unified practice among Muslims. The number of cycles in that prayer is reported variously and two different opinions have emerged. One is that this, what is called the Taraweeh prayer, is basically what is normally called Tahajjud prayers. Tahajjud prayers are a special prayer during the night which is voluntary and which usually consists of eight rakats or cycles. It could be any number of rakats, starting with two, but eight is a happy medium - It's not too much and it's not too little. Some think that's the same prayer and it's just moved to the early part of the night to make it convenient for people to gather in the mosque, rather than pray at home in the wee hours of the morning as you might have done with the Tahajjud prayer. In that case, they say it is eight rakats and there is a narrative that backs this up.
The Mother of the Believers, Aisha, may God be pleased with her, says that the Prophet, peace be upon him, used to pray eight rakats. On the other hand, some narratives seem to indicate that there were 20 rakats. That may have been because some people found that if you're just praying eight rakats and you want to finish reading the entire Quran in the whole month during these prayers, that would mean reading 1/30 parts of the Quran each night. If you read that much in just eight rakats, by the time you divide the 1/30 part into the eight cycles of prayer, it would require a long period of standing in each cycle, and people's feet will start to get tired. To make it easier, and to have the standing be a little shorter, the 20 rakats might have been introduced to stretch them out. Nowadays, the printed copies of the Quran are often arranged such that they are typeset to give you 20 pages in each of the 30 parts, for a total of about 600 pages altogether. That allows you to read one page in each cycle of the prayer, so you can finish it in 20 cycles during that special night prayer called the Taraweeh. It is a special prayer. We shouldn't be quarrelling too much about the number of cycles. If you're comfortable praying 8, do so. If you're comfortable praying 20, do so. If what's established in your local mosque is a certain number of rakats, join with them and pray with them if you can. At the end of the day, just recall that it is a voluntary prayer. If you can do it happily and for the pleasure of God, then do it. If you feel that this is a great burden, then do what you can, as much as you can. Pray some in the mosque and some at home. Keep your home alive with prayers as well, especially during the month of Ramadan, as that is the whole spirit of the Taraweeh prayer.