What are the challenges in firefighting?

The challenges in Firefighting are legion, and depend entirely on the emergency you're dealing with. A vehicle accident poses different challenges than a house fire. If I were to speak generally, I'd say the following challenges apply to almost any emergency:

1. Establishing effective command and control of the scene.

2. Maintaining a safe work environment for all involved.

3. Communicating effectively between teams and scene commander(s).

4. Effectively evaluating, requesting, and delegating tasks to resources.

That's from more of an officer's perspective. When I was inside a burning structure, my priorities changed slightly. Maintaining communication and a link with my teammate(s), monitoring the immediate environment, evaluating whether the tactics we were using were working effectively, and double checking my own safety (keeping heart rate in check, keeping my personal gear working, checking my SCBA levels, and knowing when I needed to step out and take a break) were important.