You will find many differing definitions of what Modern Art or Modernism is in the scholarship of Art History. In a nutshell, Modern Art is "A term generally used to cover art of an avant-garde (progressive) nature from the beginning of the 20th century onwards." (from The Concise Oxford Dictionary of Art Terms, 2 ed.)
Mid-to-late 19th century works are included within Modern Art pedagogy, and you will most likely start here when learning about the history and development of the avant-garde. Hence, there is no precise "beginning" to the modern art period, and it is also up for debate when modernism "ends" and the contemporary period begins. What we do know is that the modern period saw radical changes in the way art was made, thought about, and taught. You will learn about all the "isms" associated with the period and applied to movements in painting, design, photography, architecture, theater, and crafts, including Expressionism, Pictorialism, Fauvism, Surrealism, Cubism, Constructivism, Dada, Bauhaus, etc. etc. etc!
[left] Raoul Hausmann, ABCD (self-portrait), a photomontage from 1923–24, [right] Market at Minho by Sonia Delaunay, 1915.
The Bauhaus, Dessau, Germany, designed by Walter Gropius (1925-26)
Kara Walker, “Christ's Entry into Journalism,” sumi ink and collage on paper, 2017
It's hard to fully understand current art movements when there is no temporal distance between "then and now." Loosely, Contemporary Art, at least as we define it today (the early 21st century), is any art made in the recent past up until the present day, typically made by living artists or those recently deceased. As with Modern Art, the dates assigned to the origins of Contemporary Art are contested, but the period typically begins between the post-WW II era to the 1970's. When we talk about Contemporary Art we usually discuss postmodernism as the primary driving philosophy that embodies the art of today.
The Contemporary is further described as pluralistic, more inclusive of non-Western movements, and less defined by "isms," although many movements/groups such as Internet Art and Young British Artists (YBAs) have been coined by curators and art critics. For further reading, the Getty Museum has a more in-depth definition of the elements that shape Contemporary Art.
Olafur Eliasson's 'The Weather Project' at Tate Modern in 2003 (Timothy Allen)
Shirin Neshat, 'Untitled', 1995, Repetto Gallery