There are two different things in history:
dowry (from brides side to grooms side) and
bride-price (from grooms side to brides side). Also related to this is
dower (pension for widows); which wiki claims descends from dowry in Germany (though you can see the same functions being met in the code of Hammarabi thousands of years prior.).
I say "side" because sometimes it is paid by/to the groom/bride and sometimes by/to the family of them.
There is some confusion because some people, speak of dowry as any money that changes hands irrespective of direction; but that isn't strictly accurate in the formal sense. Modern translations sometimes fall into this confusion in their loose translations of Ex 22:16 which should be translated bride-price, not dowry. Bible commentaries will frequently call dowry what is more properly called bride-price.
Aside from violating a virgin (Ex 22:16), the Bible never commands dowry/bride-price; it just assumes it without comment or regulation. It was practiced widely throughout the OT. There are examples of this going to both the family and the bride at the same time (Gen 24:53, 34:12). Most of the references in the OT are to just a bride-price, but there is at least one example of probable dowry proper (Judges 1:14-15). And based on the legal history of Abraham's people, it is likely the early Israelites practiced dowry. The word translated 'marry' in the OT also implies purchase (or theft) of the bride.
But make no mistake, bride-price is commanded (Ex 22:16), a situation which occurs to probably 90% of Christian's today. That command presumes the widespread practice and that it should be done.
An interesting look at how these played out in history is the details in the
code of Hammurabi (search dowry and 'purchase price'). They practiced both, which contrasts with the tendency of ethnographers to speak of societies as practicing one or the other. It also shows a variety of uses that it was put to which are good and beneficial that make it clear it's not so much that we stopped dowry, but we replaced it with other forms.
For example, dowry was how inheritance was passed down to the daughter; today we just wait until death of the father and fight over it; if he hasn't spend it all on lavish living first. Today people are proud of spending the inheritance, but our ancestors would have found that shameful and appalling.
Dowry was also used to provide for a widow. Today we expect state to provide for widows; but in truth we all pay for it via taxes (much which is lost to administrators) or our great great grandchildren pay for it via debt spending (plus interest many times over).
Dowry was also used to provide for divorced women. Today we accomplish that by turning the husband into a slave; a rank injustice. In contrast when dowry is given the amount is fixed, known ahead of time and agreed upon; a fair and just approach that doesn't make the man a slave nor subjugate him to a bad faith, unconscionable, adhesion contract.
The bride-price also acted as a guarantee, the husband would get it back if the wife died without producing sons. This suggest ways we could use purchase price and dowry to reform our current unjust divorce system and provide restitution to men whose wives leave them in violation of their vows.
