Tahajjud, the late night prayers, or as I like to call it: the H-bomb of prayers.

I'm sure you've read about the benefits and recommendations of staying up late, or waking up late to pray. If not, here are a few:

Surat 25: Al-Furqan

63) And the (faithful) slaves of the Most Gracious (Allah) are those who walk on the earth in humility and sedateness, and when the foolish address them (with bad words) they  reply back with mild words of gentleness.

64)  And those who spend the night in worship of their Lord, prostrate and standing.

Surat 51: Al-Dhariyat

15) Verily, the Muttaqun (the pious. See V.2:2) will be in the midst of Gardens and Springs (in the Paradise),

16) Taking joy in the things which their Lord has given them. Verily, they were before this Muhsinun (good-doers. See V.2:112).

17) They used to sleep but little by night [invoking their Lord (Allah) and praying, with fear and hope].

18) And in the hours before dawn, they were (found) asking (Allah) for forgiveness.

Verily, the rising by night (for tahajjud prayer) is very hard and most potent and good for governing oneself, and most suitable for (understanding) the Word (of Allah). 
(  سورة المزمل  , Al-Muzzammil, Chapter #73, Verse #6)

And during night, prostrate yourself to Him (i.e. the offering of Maghrib and 'Isha' prayers), and glorify Him a long night through (i.e. tahajjud prayer). 
(  سورة الإنسان  , Al-Insan, Chapter #76, Verse #26)

So I am here today to tell you the whats and hows of this recommended prayer, so here's what you do:

1) You need to wake up, or plan to wake up. Waking up your family is recommended too.

Jabir reported Allah's Messenger (may peace be upon him) as saying: He who amongst you is afraid that he may not be able to get up at the end of the night should observe Witr (in the first part) and then sleep, and he who is confident of getting up and praying at night (i. e. tahajjud prayer) should observe it at the end of it, for the recitation at the end of the night is visited (by angels), and that is excellent.  (Book #004, Hadith#1651)

Also, Abu Ad-Darda has reported that the Prophet (SAW) said: "Whoever goes to his bed with the intention of getting up and praying during the night, and sleep overcomes him until the morning comes, for him will be recorded what he had intended, and his sleep will be a charity for him from his Lord." (Fiqh us Sunnah)

The best time, of course, is during the last 3rd of the night:

Abu Hurairah (RA) narrated that Allah’s Messenger (SAW) said: 'In the last third of every night our Rabb (Cherisher and Sustainer) (Allah (SWT)) descends to the lowermost heaven and says; "Who is calling Me, so that I may answer him? Who is asking Me so that may I grant him? Who is seeking forgiveness from Me so that I may forgive him?."' [Sahih al-Bukhari, Hadith Qudsi]

2) Wipe your face, brush your teeth (Narrated Hudhaifa : Whenever the Prophet got up for tahajjud prayer he used to clean his mouth (and teeth) with Siwak.  (Book #21, Hadith #237)) , look to the sky and say:

There is no God but You. glory be to You. I seek forgiveness from You for my sins, and I ask for your mercy. O Allah, increase my knowledge and let my heart not swerve after you have guided me, and bestow upon me from Yourself. You are the Granter of bounties. All praise be to Allah who restored us to life after our death and unto Him is the resurrection. 

3) Whip out a copy of the Quran and recite the last 10 Ayahs of Surah Al-Imran, starting at #190.

4) Say:

O Allah, to You belongs the praise.

You are the Light of the heavens and the earth and what is therein.

And to You belongs the praise.

You are the Sustainer of the heavens and the earth and whatever is therein.

To You belongs the praise.

You are the truth and Your promise is true.

And the meeting with You is true.

And Paradise is True.

And the Fire is true.

And the Prophets are true.

And Muhammad (SAW) is true.

And the Hour is true.

O Allah, to You have I submitted.

And in You have I believed. 

And in You have I put my trust.

And to You have I turned.

and by You I argue.

And to You do I turn for my decisions.

Forgive me my former and latter sins, and those done in private and those done in public. 

You are Allah, there is no God besides You.

5) Pray two quick Rak'ahs

Aishah says: "When the Prophet (SAW) prayed during the late night, he would begin his prayers with two quick rak'ahs." (Muslim)

Anas relates that the Prophet (SAW) said: "Prayer in my mosque is equal to ten thousand prayers (elsewhere). And prayer in the Sacred Mosque is equivalent to one hundred thousand prayers (elsewhere). And prayer in the battlefield is equivalent to one million prayers (elsewhere). And what is more than all of that is two rakahs by a servant (of Allah SWT) during the middle of the night. 

6) Pray whatever you'd like. 

There's no limit to the number of rakahs you can perform; tahajjud would count even if you just prayed one rak'ah of Witr, but whatever you pray, Witr should be the last payer. This means I'm going to have to explain what Witr is, doesn't it? 

Okay, without going in to great detail (and believe you me there is a lot of ways to do witr. In fact, I just found out I have been doing it wrong! ToT)

Right, for a 3 rukaa'd Witr, it's going to be just like any other prayer, except you don't sit for at-tahiyat in the second rukah: you just stand up again, and in the third rukah, there is a Quranic recitation after Al-Fatiha, and then there is a special dua called the Qunut to be said. So, from the beginning:

1) "Allahu Akbar"

2) Opening Dua

3) Al Fatiha

4)Quran

5) Bow

6) rise

7) Prostration 1

8) sit

9) Prostration 2

10) stand up

11) Al Fatiha

12) Quran

13) bow

14) rise

15) Prostration 1

16) sit

17) Prostration 2

18) STAND UP.

19) Al Fatiha

20) QURAN

21) Raise your hands and make takbir (say: "Allahu akbar")

22) Fold your arms like usual

23) Recite the Qunut:

"O Allah, guide me aright among those You guide aright,

and grant me health and pardon among those You grant health and pardon.

And care for me among those You care for

And bless me in what You have given

and protect me from the evil of what You have predestined.

You alone decrees, and none decrees against You.

He is not abased whom You befriend.

Blessed are You, O Lord and You are Most High.

And Allah's blessings be on (His) Prophet Muhammad (SAW)

24) Raise your hands and make takbir (say: "Allahu Akbar")

25) bow

26) rise

27) Prostration 1

28) sit

29) Prostration 2

30) sit

31) Finish the prayer (recite At-tahiyat and whatever else you normally do, then say the salams to the right and left.)

7) Recite:

Glory be to the Master, the Holy

Glory be to the Master, the Holy

Glory be to the Master, the Holy

Lord of the angels and the spirits

O Allah, I seek refuge in Your pleasure from Your anger.

And I seek refuge in Your granting well being from Your punishment.

And I seek refuge in You from You.

I cannot reckon Your praise: You are as You have praised Yourself.

8) Now that that's all done and finished with, I like to take this time to make Dua for whatever; good health, pass exams, forgiveness. If it's the last third of the night, and you did all that, I'd imagine you'd be in pretty good standing to have your prayers answered! :D