The Edge is the prologue Chapter of Mirror's Edge. This chapter must be completed before the other main storyline chapters can be accessed. The Shard is also available in the demo, and is the only mission with 2 Runner bags (one to introduce the bags, and another as a normal, secret bag).
The Edge is not really a part of the storyline of the game, though it shows when the City Protection Force became more aggressive towards Runners.
Faith Connors's task was to meet up with Celeste Wilson and give her a Runner bag. Along the way, a City Eye news helicopter spots Faith and alerts the CPF. Getting inside a building, Faith is spotted by officers that then open fire as she attempts to escape. Jumping from building to building, Faith has to take down a few City Protection Force personnel on the way to Celeste's meeting point. Meeting up with her, Faith gives the bag to Celeste, who alerts her fellow Runner of a squad of Pirandello Kruger operatives behind her. Over the radio, Mercury orders Faith to get off of the building by whatever means necessary. She is then forced to jump off the skyscraper she was on, and then manages to flee by grabbing onto the landing skid of the City Eye helicopter hovering above the edge of the building and then safely landing on another building. Mercury demands to know what Faith did to provoke the Blues to fire at her, to which she responds she did nothing. Saying he would try to get answers, Mercury then orders Faith to return back to Mercury's Hideout.