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The Pali Canon definition of a stream entrant is one who have abandoned the first three fetters namely
(There are 10 fetters, all of which must be broken before enlightenment)
So quoting from the Alagaddupama Sutta
Monks, in this Teaching that is so well proclaimed by me and is plain, open, explicit and free of patchwork, those monks who have abandoned three fetters, are all stream-enterers, no more liable to downfall, assured, and headed for full Enlightenment.
Perhaps more helpfully it's been described to me as the point at which a someone cannot fall back on the path. Forward is the only direction they will go. It is like reaching escape velocity. Until you get out of the gravity well of samsara there is always the possibly of falling back to Earth. The stream entrant no longer feels that gravitational pull so will not fall back (apologies to Newton for the analogy).
More poetically reaching stream entry has been been described as the turning about in the deepest seat of consciousness.
Regarding family life and being a stream entrant - there is no obligation to join a monastery or be a bhikku or monk. There are apparently 3000 lay stream entrants named in the Pali Canon. Sangharakshita, the founder of the Buddhist group that I belong to says that stream entry is a realistic aim for anyone seriously engaged in the spiritual life irrespective of lifestyle. Not to say that it wouldn't take a great deal of commitment of course.
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i did watch a video that make a story about buddhism. apparently someone can be a stream entrant without being a monk if it was show by a qualify monk. however , to advance further stage , one need to be a monk. this is a story about a wife who went to become a nun and then the husband later on follow her course and be a monk. so i think both of them are probably very very near during the buddha is still alive as the story said the wife actually display some sort of spiritual attainment and convince her husband to the point her husband become a monk. And apparently this happened because both of them had pray to attain the arahat fruit in one their past life during another buddha time.
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To build upon Crab's answer, stream enterer is simply one that really gets the teaching of Buddha. As in, clearly sees what Buddha meant and why: what Enlightenment is, what Nirvana is, how karma works, how suffering originates and ceases, sees how the three marks of existence are true, sees how pratitya-samutpada is true, etc. (When I say "clearly sees" I mean the direct experience of how these things manifest in the world, in front of their eyes, not just as analytical understanding of structured explanations.) Because they clearly see all this, they obviously don't have the self view anymore, have no doubts about Buddha's teaching, and clearly see to what extent the rites and rituals work, hence have no irrational clinging to them.
Now, they may not have perfectly realized (=mastered in practice) what they see yet, and that's the main difference between stream enterer and an arahant. They may still get caught up in emotions once in a while (although not in strong ones, and not for a long time), they may still have some confusions as to some subtleties of Buddha's teaching, most definitely they still have some clinging to "I" at the subconscious level, even though they don't subscribe to the self-view anymore. But in any case, because they "got it", they are not nearly as confused as they used to be before stream entry.