modern
In keeping with the present day; new and up to date in terms of technology, taste, or development.
Wir haben eine sehr moderne Küche mit den neuesten Geräten.
We have a very modern kitchen with the latest appliances.

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In keeping with the present day; new and up to date in terms of technology, taste, or development.
Wir haben eine sehr moderne Küche mit den neuesten Geräten.
We have a very modern kitchen with the latest appliances.

The adjective 'modern' is the basis for other words such as the verb 'modernisieren' (to modernize something) and the noun 'die Modernisierung' (modernization). The name of the art and cultural era 'die Moderne' (modernism/the modern era) is also derived from it.
Wir wollen unsere alte Wohnung komplett modernisieren.
We want to completely modernize our old apartment.
In the meaning 'zeitgemäß' (contemporary/up to date), 'modern' is often used to describe technology, design, or architecture. Typical combinations are 'moderne Technik' (modern technology), 'moderne Möbel' (modern furniture), or 'ein modernes Gebäude' (a modern building).
Das neue Bürogebäude hat eine sehr moderne Fassade aus Glas und Stahl.
The new office building has a very modern facade made of glass and steel.
When 'modern' is used in connection with art, literature, or music, it often refers to the art-historical era of 'die Moderne' (modernism). Watch for nouns like 'Kunst' (art), 'Malerei' (painting), or 'Architektur' (architecture).
Das Museum stellt viele Werke der klassischen Moderne aus.
The museum exhibits many works of classical modernism.
When it is about ways of thinking, 'modern' describes a progressive or open-minded attitude. Common combinations are 'moderne Ansichten haben' (to have modern views), 'modern denken' (to think in a modern way), or 'eine moderne Familie sein' (to be a modern family).
Er hat für sein Alter sehr moderne Ansichten zur Gleichberechtigung.
For his age, he has very modern views on equality.
new
'neu' and 'modern' can both describe something that does not seem old. 'neu' mainly says that something was created, bought, or changed recently. 'modern' says that something fits the present day, for example in technology, design, or style. 'Ein neues Handy' is a phone that has just been bought or made. 'Ein modernes Handy' has up-to-date technology. Something can be new but not modern if it is unused but seems old-fashioned. And something modern does not have to have been bought brand-new.

Die Wohnung ist modern eingerichtet und hat eine offene Küche.
The apartment is furnished in a modern style and has an open-plan kitchen.
Am Montag hat Hannah einen neuen Laptop für die Arbeit bekommen.
On Monday, Hannah got a new laptop for work.