Felix Mendelssohn Bartholdy, born in 1809, was a german romantic composer and pianist. He is one of the most played and best known german composers, conductos and pianists of all time. Listen to his “Sommernachtstraum”, in english Midsummer Night’s Dream, played by the Gewandhausorchester Leipzig:
And here’s a short nice remix:
World Music - the Music Journey 
Nice piece of music. But I wouldn’t say Mendelssohn was mostly known in his role as a piano player. More as a composer, conductor and music teacher.
So he was the founder of the first german musical conservatory in Leipzig and the conductor of the Gewandhausorchester playing in your youtube video ;)
Thanks so much! The second information was new to me! Will edit the first a bit ;)