These answers are really making me feel young, thanks guys!
My first was Bon Jovi on the Slippery When Wet tour. July 21, 1987 - Washington DC
I was definitely a teen of the late eighties/early nineties with most of my concerts. U2, Def Leppard, Pearl Jam, Nine Inch Nails, etc. Blink 182, Offspring, Green Day, etc.
My most recent one was Weird Al.
I've gotta say that I'm not actually quite nerdy enough for a Weird Al concert, but it was still a lot of fun as a concert.
One of my most memorable was Simon and Garfunkel. I felt like I was stepping back in time on that concert from them and they really blew it out of the water.