They are from the Garde républicaine
About their Duty:
They (and their Infantry counterpart) are to protect the President during Parades (14 juillet and other), the high ranking people who come visit the President in Paris, the Presidential palaces (Palais de l'Elysée) , The National places, (the 2 Houses of our Parliament: Assemblée et Sénat) the Ministers Offices.
About their uniform: I was told they wear a Dragoon uniform that goes back from the early beginning of the third Rebublic. After Napoleon the third collapse after the Franco-Prussian war in 1870.
Hope this help.