Tahajjud (qiyam al-layl) is a sunnah prayer performed during the night after Isha has already been prayed. It consists of one or more prayers, two rakaat each. It is usually prayed in the last third of the night.
Tahajjud is the best of the Sunnah prayers. The Messenger of Allah ﷺ is reported to say:
'Perform the night prayer, for it is the practice of the righteous before you, a means of bringing you closer to your Lord, an expiation of bad deeds, and a protection against sin.' (Mishkat al-Masabih, 1227).
Tahajjud is also a means to be included among the people of remembrance. In a hadith reported from Abu Huraira, the Messenger of Allah ﷺ said:
'He who wakes up at night and wakes his wife, and they pray two rakaat together, will be recorded among those who remember Allah often.' (Abu Dawood, 1451).
The last third of the night is the best time to make sincere dua (supplication).
Allah says about the last third of the night in a hadith qudsi:
"Is there anyone to invoke Me, so that I may respond to invocation? Is there anyone to ask Me, so that I may grant him his request? Is there anyone seeking My forgiveness, so that I may forgive him?" (Bukhari, 1145).
A simple way to start Tahajjud:
Wake up 20 mins before Fajr
Perform Wudu (ablution)
Pray at least two rakaat and make dua
You can recite any portion of the Quran you are comfortable with. Once you are used to it, you can wake up earlier and pray additional prayers.
Remember, Tahajjud is a sunnah prayer. Fard (obligatory) prayers are more important than sunnah, so establish fard prayers first.