Sonic Heroes was the first game I fully completed where my brother had it on PS2. I did have a Spyro game but was too stupid to ever finish it, having got my mum at the time to do a lot of the levels for me.
For some reason MW2 wouldn't appear so I chose the next best thing. I also played a lot of MW3, and then later BO3.
My brother had Need for Speed: Most Wanted which was great, but for me Carbon will always have a special place in my heart. I remember getting it for my birthday and I absolutely loved the story and the sheer amount of customisation available on cars, especially with bodykits and JDM culture (which seems to be absent even from big titles like FH6 nowadays). Underground 2 was also great.
I used to play TF2 normally (if there was ever such a thing) and then transitioned to community servers and did a lot of minigames stuff, with endless nights of trying not get killed by a massive Saxton Hale.
I remember playing GTA: Vice City around a friend's house and then begging to get GTA IV when it came out. I still reminisce over how dark and gritty the theme of the game was. Although GTA V is obviously fantastic, I remember many a-school night rotating between CoD and GTA IV.
DarkOrbit was the first MMO I ever played and to be honest, I'd probably still play it, but having tried it again not long ago the game has absolutely lost its core appeal and all the successive updates has made the gameplay suck. SWTOR was next on the MMO list - tried WoW and RuneScape, never really got into them. Still need to continue with Guild Wars 2.
Garry's Mod is obviously a given through PERPHeads and SimplyPERP. I also played HL2:RP and S.T.A.L.K.E.R. RP on the Clockwork schemas for years, but also loved all the minigame stuff like TTT and one where you had to build a bobsleigh. I got into Minecraft watching Shadow of Israphel from the YogsCast, using a cracked client for literal years before actually buying a key.
CS:GO was obviously huge for me as well. Sadly, I never got to try the 1.6 days or before.