Tadabbur is an Arabic word which means -To ponder, reflect and think.
Literally, it means to look into the consequence or the end result of the matters, e. g. if we did that what would happen to us.
Tadabbur, means pondering over the meaning of the verses of the Qur’an and contemplating them.
It is not enough to learn the Qur’an and recite it profusely. But what is required from us is to understand the words and the meaning of the aayaath. To know the matter (or the case or what the ayath says) and this cannot be attained except by reading the Tafseer (the interpretation), by heart or heedful mind and not by ears and eyes.
Tadabbur is comprehensive and deep thinking and consideration in order to reach to a guidance or indications.
For example, the word Rahmah (Mercy) indicates many things such as abundant knowledge, abundant provision (if Allah (Subhaanahu Wa Ta’aala) bestowed upon you abundant provision this indicates a mercy from Him), if your manners have been refined and perfected by the Qur’an, then this is a mercy from Allah, you heard a story and took the admonition then this is a mercy, and so on… This is when you just think considerably in the word Rahmah (Mercy).
Accordingly, we say Tadabbur means the utmost understanding of the meaning of the words of Allah by reading the Tafseer, deep thinking and consideration over the meanings of the verse, accept its guidance fully and completely. Mould yourself according to its guidance. Take each passage of the revelation as addressed to you. Put yourself in the same state (similar situation) [i.e. ask yourself whether you have gone through a similar situation or not].
After that comes acting upon it, putting it into practice and that would be by developing an inner response to the Ayah, and express it by praising Allah, seeking His forgiveness, etc. Kitab Al Fawa’id by Imam Ibn Al Qayyim
Bear in mind that Tadabbur, pondering over the meanings of the aayaath of the Qur’an, is obligatory upon every Muslim reciting the Qur’an.
The Qur’an itself emphasizes that mere reading or reciting of it is insufficient. One needs to reflect upon what one reads, and then act upon it. Allah said: (This is) a Book (the Quran) which We have sent down to you, full of blessings that they may ponder over its aayaath, and that men of understanding may remember. Qur’an Surah Saad 38:29 The Word used in the ayath is YADDABBARU.
كِتَابٌ أَنزَلْنَاهُ إِلَيْكَ مُبَارَكٌ لِّيَدَّبَّرُوا آيَاتِهِ وَلِيَتَذَكَّرَ أُوْلُوا الْأَلْبَابِ
Allah also said: Then do they not reflect upon the Qur'an, or are there locks upon [their] hearts? Qur’an Surah Muhammad 47:24
Allah said in Qur’an: “Do they not then consider the Qur’an carefully? Had it been from other than Allah, they would surely have found therein much contradictions.” Qur’an Surah an-Nisa’ 4:82
أَفَلاَ يَتَدَبَّرُونَ الْقُرْآنَ وَلَوْ كَانَ مِنْ عِندِ غَيْرِ اللّهِ لَوَجَدُواْ فِيهِ اخْتِلاَفاً كَثِيراً
إِنَّمَا الْمُؤْمِنُونَ الَّذِينَ إِذَا ذُكِرَ اللّهُ وَجِلَتْ قُلُوبُهُمْ وَإِذَا تُلِيَتْ عَلَيْهِمْ آيَاتُهُ زَادَتْهُمْ إِيمَاناً وَعَلَى رَبِّهِمْ يَتَوَكَّلُونَ
“The believers are only those who, when Allah is mentioned, feel a fear in their hearts and when His Verses (this Qur’an) are recited unto them, they (i.e. the Verses) increase their Faith; and they put their trust in their Lord (Alone).” Qur’an Surah al-Anfal 8:2
Another benefit that one can get from the Tadabbur is that the person will reach to the rank of certitude and knowledge that the Qur’an is the Word of Allah, because he will see that all the parts of the Qur’an affirm each other and agrees with each other. All the stories and news that are mentioned are repeated in many parts of the Qur’an, and all of them are confirming and agreeing with each other, there is no contradiction between them. Therefore, Allah (Subhaanahu Wa Ta’aala) said: “Had it been from other than Allah, they would surely have found therein much contradictions.” Qur’an Surah Muminoon 23:68
Therefore, Allah (Subhaanahu Wa Ta’aala) commanded and encouraged us to make the Tadabbur. He (Subhaanahu Wa Ta’aala) informed us that the Tadabbur is the purpose behind revealing the Qur’an, and it is one of the conditions required for the recitation of the Qur’an. Accordingly the breasts will be opened and the hearts will be enlightened and it reaches with the slave to the degree of certainty.” [Tayseer al-Kareem]
Ibn al-Qayyim (Rahimahullaah) pondered over the following ayath ( Quran Suarh Al-Baqarah 2:216), he came up with seven benefits and wisdoms out of it::
وَعَسَى أَن تَكْرَهُواْ شَيْئاً وَهُوَ خَيْرٌ لَّكُمْ وَعَسَى أَن تُحِبُّواْ شَيْئاً وَهُوَ شَرٌّ لَّكُمْ وَاللّهُ يَعْلَمُ وَأَنتُمْ لاَ تَعْلَمُونَ
“And it may be that you dislike a thing which is good for you and that you like a thing which is bad for you. Allah knows but you do not know.”
He concluded all these actions of the heart:
a. إيمان وتصديق, belief in Allah’s Word, to believe that Allah knows and I don’t know.
b. Complete Tawheed , full authorization to Allah in all one’s affairs.
c. Submission to Allah (Subhaanahu Wa Ta’aala).
d. Tawakkul, Complete reliance and putting trust in Allah. Whatever Allah (Subhaanahu Wa Ta’aala) has decreed for you is the best.
e. Inaabah, to turn unto Allah in all our affairs and seek His Guidance for every matter.
f. Rida, to be pleased and satisfied with what Allah (Subhaanahu Wa Ta’aala) has chosen for you [of husband, children, job, nationality, country, place, house, etc.]
g. Hamd and Shukr, thanking Allah (Subhaanahu Wa Ta’aala) for the blessing that He has bestowed upon you.
h. invoke Allah (Subhaanahu Wa Ta’aala) to keep you firm when you face any misfortune or calamity, and to grant you also patience.
Eight acts of the heart, one can perform while sitting and pondering over this ayath.
In this way, you have acted upon this verse. If you recited the Qur’an without Tadabbur, then you will be deprived from getting the fruit. The Qur’an should change our behavior and characters. 
Indeed, there are many scientific facts mentioned in the Quran which was unknown to the world until very recently. So it is beyond doubt one of the miracles of the Quran. As Allah says in Surah Fussilat (41:53)(the meaning of which can be translated as):
“We will show them Our signs In the universe, and In their ownselves, until it becomes manifest to them that This (the Quran) is the truth. is it not Sufficient In regard to Your Lord that He is a witness over All things?”
But while trying to prove the scientific facts from the Quran the following things are to be kept in mind:
 Kitab Al Fawa’id by Imam Ibn Al Qayyim, http://www.4theseekeroftruth.com/?p=203