A dowry is the money, goods, or estate that a woman brings to her husband in marriage.
The giving of dowries is an extremely old practice that is thought to predate writing. While not very common in this time period this practice still exists in several places. Similar practices exist that are not called dowry. Take for example in the USA it is customary for the bride's family to pay for the wedding; some people view this as being a dowry.
A country that still has the practice of giving dowries is Japan. It is not uncommon for the parents to give a dowry of some type. A famous example would be Princess Sayako who in 2005 left the Japanese royal family and becomes a commoner through marriage. The Japanese government gave Princess Sayako 152.5 million yen upon her marriage as a dowry (the legal limit according to imperial law).
I have made some attempt to learn about dowries in Japan with little success (though there is also a similar practice called Yuinou (or Yuino) where the two families exchange betrothal gifts). The reason I have attempted to find out info on them is because a couple characters in Ranma 1/2 have dowries most importantly Ukyo. Some people claim that Ukyo's engagement is invalid because she gambled away her dowry prior to Ranma's father stealing it. The thing is a dowry is not needed or required for an engagement/marriage. Having one and the size of the dowry does show how wealthy/important the girl is but it really does not matter for the engagements/marriages at all. A dowry does not really go to the guy/his family. What a dowry really is is the girls inheritance meant for taking care of her and her children. The girl generally becomes part of the new family (transferred from her family registry to his) and it is a means of giving her inheritance to her. From what I read in the case of the girl dying without having children or divorce it's customary to give the dowry back to her family. Of course I can find very little on modern dowries except that they still exist even though they are of less importance. Most of what I found was about them in Edo Japan (if I find out more I will likely edit this post).