Please see the picture below. What is this 'box', gilded perhaps, that can be found in Catholic churches? Some make a genuflection before it, and even the Pope bows reverentially before it. What is its history and significance? What's kept in it? What is the name and purpose of the lamp beside it? Does the the 'box' have any scriptural basis? In the answer, please make a reference to the Church law that governs this box.
It is the tabernacle where the Eucharist is reserved for adoration. I personally do not like above picture because the Church in question looks very modern and plain.
Here are some better pictures
The tabernacle is a liturgical furnishing used to house the Eucharist outside of Mass. This provides a location where the Eucharist can be kept for the adoration of the faithful [hence the genuflecting and the Pope's reverential bow before it] and for later use (e.g., distribution to the sick).
It also helps prevent the profanation of the Eucharist. Thus the law requires, "The tabernacle in which the Eucharist is regularly reserved is to be immovable, made of solid or opaque material, and locked so that the danger of profanation may be entirely avoided" (CIC 938 §3).
The word tabernacle means "dwelling place." Any place someone dwells is a tabernacle. The term is also sometimes used for a temporary dwelling place. Thus the tent-like sanctuary that the Jews used before the Temple was built was called the Tabernacle, because God dwelt there. Similarly, for the feast of Sukkot the Hebrews erected temporary shelters to live in for the festival, which is often called "the feast of tabernacles" or "the feast of booths" as a result.
The tabernacle in Church is so named because it is a place where Christ dwells in the Eucharist.
As regards the lamp beside the tabernacle, it is a sanctuary lamp:
A wax candle, generally in a red glass container, kept burning day and night wherever the Blessed Sacrament is reserved in Catholic churches or chapels. It is an emblem of Christ's abiding love and a reminder to the faithful to respond with loving adoration in return.
Please note: The sanctuary lamp also serves to inform people that the Blessed Sacrament is indeed present in the tabernacle, because on certain occasions the tabernacle is empty and then it would be inappropriate to genuflect to it (though one might genuflect or at least bow to the altar on which it rests or to a nearby crucifix). – Andreas Blass
Please see also: Tabernacle | New Advent.