Question: What does sadaqah jariyah mean?
ANSWER
It is a kind of sadaqah that brings thawab to our books of deeds even after we die. It includes any good deed that benefits humans, such as building a mosque, fountain, or road, planting a tree, leaving a beneficial scholarly work. It is stated in a hadith-i sharif: 

(When a person dies, his deeds come to an end except three: sadaqah jariyah, leaving a beneficial scholarly work, or a pious child who will make du’a and seek forgiveness [istighfar] for him.) [Muslim]

Not everybody has the means to get a mosque done or to write a scholarly book. But there is an easy way. Contributing to the spread of a beneficial book earns a person the thawab of actually writing that book. It is stated in a hadith-i sharif:
(If a Believer learns ‘ilm and spreads it, its thawab continues even after his death.) [Ibn Majah]

What is meant by a beneficial work is any work that is beneficial to our worldly life or the life to come. Books of fiqh, tafsir, medicine, physics, and chemistry are of beneficial books. Cassettes, CDs, films, radio and television programs, newspapers, magazines, websites and any broadcast are included in sadaqah jariyah, on condition that they are beneficial. It is stated in hadith-i sharifs:

(There is the reward of a hundred martyrs for the one who revives a sunnat of mine.) [Hakim] (Then think the magnitude of the thawab of reviving a fard or wajib duty.)

Those who conform to true Islamic books written by Ahl-i Sunnat scholars will earn the thawab of hundreds of martyrs. Abiding by sunnat and reviving it can be done by reading the books of Ahl-i Sunnat scholars.

If one gives a book of Ahl-i Sunnat scholars, such as Ethics of Islam, as a gift on the days of, say, our religious festivals, one will earn the thawab of a hundred martyrs. By doing so, one spreads sunnat deeds as well as fard and wajib ones. Giving a corrupt book, on the contrary, is something that makes the donor share in the sins committed.

Hadrat Muhammad Masum stated:
The thawab of sadaqah [and the thawab of reading the Qur’an al-karim] should first be presented to the blessed soul of our Master the Prophet and then should be presented to the souls of the dead. Thus, there is more hope of acceptance. It is better to present the thawab to the souls of all Believers. The total thawab reaches each Believer without any decrease in the thawab of the donor. (Vol. 2, Letter 36)

Question: My father said, “Take this money and donate it to a mosque.” Is it permissible to give this money to institutions where amr-i maruf is performed by spreading the books of Ahl-i sunnat scholars?
ANSWER
Of course, it is permissible. It is given to spread Islam, which is more important.

Question: If a small mosque is demolished and a big mosque is built by the donations of someone else, does the thawab of the one who made donations for the construction of the former come to an end?
ANSWER
No, it does not.