"Modern art" is a vague historical term for the period between the late 19th century to around ~1970 in the Western canon of fine art. It refers to the general trend of art that directly challenged historical and traditional modes of thinking and representation. Dozens of diverse art scenes and movements, over the span of a century, can be wholly addressed as "modern art".

"Modern art" is often confused with "contemporary art" because they appear to mean the same thing, however they actually refer to two different historical periods of art. It seems every expert has their own timeline of where modern art ends and contemporary art begins, or their own slightly different interpretation of what the term means, but a safe definition is that contemporary art is work made within the last thirty to forty years.

If we use the literal definition of the word contemporary (meaning "at the same time"), then all art is considered "contemporary art" at the time it was made. If you lived a century ago, Picasso and Duchamp would be contemporary art. If you lived a thousand years ago, that art would be contemporary to you, and so on.

Some art scenes and movements declare their own terms for their work, for example as the Surrealists did in a manifesto titled Manifeste du surréalisme. Some are given terms by art critics promoting the work, for example Neo-Impressionism or Pop Art. It is possible that some art historian a hundred years from now will devise a new catch-all term for the art of the late 20th and 21st centuries, just as the term "Renaissance" was retroactively applied in the 19th century.

To add to u/153x153's answer (a week late, but this is my jam so I couldn't pass it up):

Most art historians classify modern art in its broadest sense as art that embraces the properties of the medium.

We see modern art begin in the 19th century with impressionist and post-impressionist painters who, while inspired by Japanese ukiyo-e, embraced the inherent flatness of the canvas. These prints can be seen in the background of Van Gogh's Pere Tanguy and their influence is most obvious in Almond Blossoms. This flatness can also be seen in Manet's Olympia. These types of work contrast greatly with the type of paintings that peaked during the Renaissance that attempted to mimic nature as much as possible and create a "window" into the work's reality.

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Cezanne is considered the father of modern art for this reason. His use of skewed perspectives exaggerated the presence of the canvas rather than try to create an illusion of the lack of canvas. Cezanne didn't care about the accurately reproducing reality and his work set the tone for Fauvism and Cubism in the early 20th century.