Well, to make a long story short, Felix Mendelssohn was not Jewish! he was born Jewish, but he later on converted to Christianity, which makes him not Jewish.
Since Felix Mendelssohn's mother was Jewish, and in Orthodox Judaism and Conservative Judaism, those with a Jewish mother (or those who underwent a halakhic conversion) are considered Jewish, then that makes him Jewish (though once again, he was later on converted to Christianity, which makes him not Jewish after all). His father was Jewish too.
Felix Mendelssohn was born on 3 February 1809 in Hamburg. Felix Mendelssohn passed away on 4 November, 1847 (aged 38).
Felix was first brought up without religious education, and was baptised by a Reformed Church minister in 1816, at which time Felix was given the additional names Jakob Ludwig. His full name was then Jakob Ludwig Felix Mendelssohn Bartholdy.
He wrote Overture to a Midsummer Night's Dream and founded the Leipzig Conservatory of Music in 1843.
If you see any details that require update, please Let us know!