To work on or toward goal or task with great determination, diligence, and vigor. I've been forging away at my little business here for two years, and now I've really established myself within the community. The team has been forging away at their shot for a championship all year.

To proceed with something. The start of my presentation did not go as I had hoped, but I had to just forge ahead and keep talking.

To proceed with something. The start of my presentation did not go as I had hoped, but I had to just forge on and keep talking.