This terminology isn't specific to Russian language. It's rather a common point for the whole Slavic culture.
Замужество ultimately means a status of a married woman who lives at her husband's place (see patrilocal residence). Note that living at wife's house was considered improper, so there even exists a specific term "примак".
On the other hand, ...
We have more words related to the topic.Брак (wedlock,marriage ) is an official term.The verb is вступить в брак. It is related to both parties (a man and a woman ) Супружество (spousehood ) is the state of being married.
ЖЕНИ́ТЬСЯ, женюсь, женишься; совер. и несовер. perfective and imperfective.
1. на ком. О мужчине: вступить (-пать) в брак of a man: ...
The existing answers provide a wealth of interesting information. But the questioner is a beginner and so needs a simple guide to talking about marriage and weddings. Here is my attempt:
The Most Important Words About Weddings
брак--the institution of marriage; a specific marriage
вступить в брак--to get married; literally "to enter into marriage", ...
Cвадьба is a ceremony of getting married, better translation would be wedding. If you mean marriage you could use word брак without concern of direction.
Also женитьба describes matchmaking from man's perspective — a man getting a wife(жена — женитьба). And замужество is a state of being married, from a woman's perspective — wife behind a husband за мужем.
When speaking about a married couple as a whole (in plural) "они поженились", or even "они женаты" can be used, even though, strictly speaking, only the man "поженился/женат", the woman "вышла замуж/замужем". Can't think of any common singular synonym that can be used for both sexes. In more official language, you could say "состоит в браке" or "вступил[а] в ...
Well, there's official and very pompous "сочетаться браком" or even "сочетаться узами брака" but no one sane will use it in casual speech. "Пожениться" is indeed the form that can be used in general.
Other words worth to know:
расписаться (here - зарегистрировать в официальном учреждении свой брак с кем-либо )
бракосочетаться (almost as bad as "...
Just to add, "женитьба" is a noun formed from the verb "жениться", which comes from the word "жена=wife"and literally translates as something like "to get wifed".
"Замужество" is formed from the phrase "выходить замуж", literally translates as "to go after/behind a husband", "муж" being "husband".