My favorite moments in PBW games are when you have been building up, have a nice fleet, and then go to war with your neighbour who is roughly equally strong with an equal number of ships. I love the anticipation of the next turn coming in with the battle reports and seeing that I won by better anticipating the enemy ship designs.
My best moment was the first time this happened, was also my first PBW game, I had stockpiled some 800k in minerals, neighbour (friend)suddenly declares war, I knew hu uses PPB's, I start researching phased shields like nuts and spent 700k in two turns retrofitting all my fleets...then we battle (about 50 vs 50 ships)and: He lost all, I lost about 6 ships
3 turns later he surrendered.
I couldn't stop smiling all night :-)