If follow a strictly design/art history viewpoint:

Modern
Refers to designs from a specific time period and source location in design and art, specifically the early 20th century (1900 to 1950/70) when the Modernism movement in Western art culture reached its peak. Design sensibility tends to favour very straight, angular shapes, with an emphasis on functionality over form, and integration with nature/outdoor design. It's a pushback against previous era of Art Deco and Victorian/Edwardian design ("Classic" design) which is very fancy and full of ornaments and symmetry. At the same time it's also a pushback against how at the time industrialisation made many people stuck at home/office and have "lost" connection to the outdoors. So design wise Modern design tends to favour neutral colors but warm tones and incorporate wood/natural stone, as well as deliberate asymmetry. There's also a sense of how outdoor type materials (wood and cement) and elements like big glass windows and open entrances is mixed into the indoors to give the illusion that outdoor and indoor are one big open living space.

Minimalist
It's a philosophy that's not directly tied to design. It's a mindset that emphasis on living with purpose, with no materialistic tendencies or the need to get anything "extra". Design wise it's very zero ornaments, everything is clean and neat. If it doesn't have a specific function then it is removed entirely (like decorations such as flower vases and paintings).

Contemporary
It has no specific reference or philosophy, but refers to what is trendy and current. It's the most diverse and most customisable meaning because it refers to what's in trend and favoured, depending on the time period and culture. It can incorporate and mix many elements from multiple design styles. It's also the one that will go "outdated" fastest what's contemporary 20 years ago would not be contemporary now.

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Note that this is the perspective from art history, but as a client it is more important that you choose based on your personal taste. Modern-minimalist-contemporary has shared elements and so are often blended.

This post has been edited by DragonReine: Dec 16 2020, 06:12 PM