Here are some tutorials for using the A* Pathfinding System:
Firstly, make sure you have read the Get Started Guide in the docs
http://arongranberg.com/astar/docs/getstarted.html
A* Pathfinding for RTS or Tower Defense Style Games
quill18 has created an excellent tutorial for getting started with the A* Pathfinding Project. It explains a lot of stuff the above get started tutorial does not cover, so check it out: http://youtu.be/bkCLqPpopvE
It is not super up to date (posted in 2013), but I think the tutorial is still useful.
Using the A* Pathfinding Project with AI Behaviors
AI Behaviors package in the forums:
http://forum.unity3d.com/threads/133180-AI-Behaviors-Made-Easy!-(Walker-Boys)
Make A 2D Game With Unity3D Using Only Free Tools
The tutorial uses a quite old version of the A* Pathfinding Project, but it is still relevant even if not everything will match up with the current version of the system.
http://www.rocket5studios.com/tutorials/make-a-2d-game-with-unity3d-using-only-free-tools-beginning-enemy-ai-with-a-pathfinding/