A lot of good ones mentioned already. I'll name some:
The ending scene to the first Fallout. "You're a hero...and you have to leave".
Hakunin's death and discovering the fate of Arroyo in Fallout 2. Having gone through the journey to get that GECK, only to come back and realized my tribe was kidnapped, I felt dedicated to getting them back. It was quite the moment when first playing.
The entirety of Dead Money, from its tragic backstory, the events with Elijah, all of the characters, and the ultimate message behind it all. I will always mention this, because its my all time favorite experience throughout the series.
There is one moment in Fallout 4 that really got to me (so far), but giving Marlene's holotape to Arlen Glass and hearing his entire story for constructing the Giddyup Buttercup toys. Sad, yet a touching moment. In the end, you feel like you gave him the peace he had always wanted for so many years.
Ah, one of my favorite moments in Fallout 2 when realizing the ultimate effect of such a small thing. My jaw dropped when realizing I actually did that.