das Eigenheim
own home
A house or apartment that one owns and usually also lives in oneself.
Viele junge Familien träumen davon, irgendwann ein eigenes Eigenheim zu besitzen.
Many young families dream of one day owning their own home.

own home
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own home
A house or apartment that one owns and usually also lives in oneself.
Viele junge Familien träumen davon, irgendwann ein eigenes Eigenheim zu besitzen.
Many young families dream of one day owning their own home.

The word is made up of 'eigen' (own) and 'das Heim' (a poetic word for home). So the meaning is literally 'an own home' or 'a home of one's own'.
Der Bau eines Eigenheims ist ein großes Projekt.
Building your own home is a major project.
Common verbs used with 'Eigenheim' (own home) are 'ein Eigenheim bauen' (to build an own home), 'kaufen' (to buy), 'finanzieren' (to finance), or 'besitzen' (to own). A very common phrase is also 'der Traum vom Eigenheim' (the dream of owning a home).
Sie haben lange gespart, um sich den Traum vom Eigenheim zu erfüllen.
They saved for a long time to fulfill their dream of owning a home.
'Eigenheim' (own home) is a common term, but it is often used in more formal contexts such as banks, insurance companies, or the news. It sounds more official than 'eigenes Haus' (one's own house).
Die Regierung fördert den Kauf von Eigenheimen durch günstige Kredite.
The government is promoting the purchase of homes through low-interest loans.
house
'das Haus' and 'das Eigenheim' can both refer to a place to live. 'Haus' mainly describes the building itself. It does not say anything yet about who owns it. 'Eigenheim' emphasizes that you own the place where you live and usually also live there yourself. Learners may confuse the words because an Eigenheim is often a house. But an apartment you own can also be an Eigenheim. When ownership is important, 'Eigenheim' fits better; when only the building is meant, 'Haus' is more natural.

Sie sparen seit Jahren für ein Eigenheim am Stadtrand.
They have been saving for years for their own home on the outskirts of town.
Das Haus gegenüber hat einen großen Garten und rote Fensterläden.
The house across the street has a large garden and red shutters.