Since Destiny 2 has gone free to play it looks as if the requirements to unlock Flashpoints has changed.
Originally these would be unlocked by progressing to a certain point in the main story, but as of New Light, you now need to reach 900 light with a character and this will then unlock the weekly challenges.
Flashpoints involve completing various activities (public events, lost sectors, adventures etc) on a target planet which changes weekly. Once enough of these have been completed you can then turn them in to the planet's vendor. To find out which planet is currently "hosting" a flashpoint, visit:
https://www.bungie.net/en/Explore/Detail/Event/463010297