Taraweeh is a special component of Ramadan. It is a special name given to voluntary night worship in Ramadan which is mentioned in Surah 73 of the Holy Quran, with the order given to the Prophet Muhammad (peace be upon him) and the Muslim community to stand up in prayer for most of the night, or one half, or one third of it. The order is relaxed in the last verse of the surah to make it voluntary, rather than obligatory.
These days debates are going on about the number of raka'ats to be prayed in Taraweeh. There is consensus among the ulema that it can be either eight or 20. There are indications to both in several ahadith. We should not get into arguments over the number of raka'ats as Taraweeh in itself is not obligatory and no sin is incurred if it is not offered. But it is highly recommended because any act of worship entails maximum rewards in Ramadan. Taraweeh is a means to achieving a higher standard of spirituality.
It is also all right for anyone to offer Taraweeh prayers at home, individually, or in a group. But when there is an arrangement made in your local mosque, it would be more appropriate to pray in a congregation.
The Prophet (PBUH) offered the Taraweeh prayers in his mosque the first night when he was joined by one or two people, and on the following night he was joined by a fair-sized congregation. On the third night, he looked through his door and found the mosque full of people. Therefore, he did not come out. When asked why, he said that he did not wish that this prayer should become obligatory. This shows how thoughtful of his community the Prophet (PBUH) was. Even in matters of worship, he always wanted what was easier for them. If he were to offer this prayer in the mosque every night, throughout Ramadan, people would over the years elevate it to the obligatory or semi-obligatory status.
Taraweeh continued to be offered in congregation, but without regular arrangements. What Umar did was to organise it in a proper way. One night in Ramadan he came into the mosque and found several groups of worshippers offering the Taraweeh prayers in several congregations. He disliked this, because it suggested division within the Muslim community. Therefore, he told them to form one congregation and appointed Ubayy ibn Kaab to lead the congregation. Ubayy was one of the best reciters of the Holy Quran among the Prophet's (PBUH) companions.
Tahajjud is a voluntary prayer which may be offered at any time during the night, after one has offered the obligatory prayer of Isha', and before offering dawn prayer, or Fajr. However, the latter part of the night is preferable for this prayer. It can be offered any night, and one may pray short or long raka'ats, as one wishes. If one is praying in congregation, it is recommended to the imam to make sure that everyone in the congregation is willing to have long raka'ats, before one starts. If one is offering night worship, or Tahajjud, one is recommended to finish it with witr. The minimum number of rak'ah of Tahajjud is two, excluding witr, and there is no maximum. However, it is authentically reported that the maximum number of raka'ats the Prophet (PBUH) offered on any one night was 11, including witr. It is suggested that the maximum he offered was 13.
"Narrated Ayesha that the Prophet (PBUH), in Ramadan or otherwise, (usually) did not offer more than 11 raka'ats in Tahajjud. First he offered four raka'ats and what to say of their magnificence and extent. Then he again offered four raka'ats and do not ask of their magnificence and extent. Then he offered three witrs". (Bukhari, Muslim)
One may recite the Holy Quran loudly or may read it privately in Tahajjud. It is preferable to recite loudly if one is alone or if one is in congregation, or if there are people nearby who would like to listen to, or may benefit by the recitation.
On the other hand, if there is someone close who is doing his own night worship, or someone who is asleep and may be disturbed, then it is preferable to read the Holy Quran privately.