I mostly play multi-player games if there is a co-operation mode, with real life friends or family members (I used to play the original Halo on my XBOX with my father and my cousin from time to time). The "real" multi-player games I absolutely adored were the off-line games of the past in which we could gather as friends together at someone's place and play games like Goldeneye 007, Perfect Dark, Twisted Metal, various racing games, in split-screen mode. Additionally, I used to take my own PC (back in around 2004 or 2005) to my cousin's place and we connected our PCs in a LAN to play Unreal Tournament 2004 and Command & Conquer games (amongst others) together.
Basically what I love about multi-player (off-line in split-screen, in LAN mode or on-line) really is co-operating with friends for a common objective, since I'm not really a fan of competition against one another even if it's just "for fun". Although with rare exceptions such as Team Fortress 2 since the game itself (the themes, arty style, animations, characters and their personalities, etc.) swims in a light and comical atmosphere that I do appreciate every now and then. I really don't give a rat's ass about hats though, they don't bother me, and some of them are funny anyway especially on the Heavy Weapons Guy.
Other aspects of multi-player such as unpredictability (if the A.I is absent and only playing with or against humans) or "skills" that can improve your performance over time, unlocks (eeew, really), etc., I am indifferent to most of them. The only one that does sometime annoy me are the unlocks yeah, since of course I rarely have enough time to play enough of the game to reach "medium" or even lesser ranks or positions or whatever that would allow me to gain access to "more powerful" or "more balanced" weapons or other types of items, that of course others around will always have when you don't. It's not that unlocks really affects the games that have that feature though, most unlocks really don't give much advantages if any. But I am aware that some unlocks in some games do really give an edge in certain circumstances, such as a sniper rifle being more stable when zooming further or being more precise or with a higher firing rate.
Generally speaking I rarely play multi-player focused games that don't have objectives/quests that I can complete with a friend or players of the same faction (for instance in an MMORPG, especially those when I can chose to PvP or not), as I said with just a few exceptions here and there however. Some of the best co-operation games I've played and hold a place in my heart to this day are 8-bit or 16-bit era Beat Em' Up side scrollers, Twisted Metal/Vigilante 8 (PS1, N64), Goldeneye 007, Perfect Dark, Halo series (especially the original, even to this day is still fun), Left 4 Dead (with friends I know in real life mostly, or with players of a community I used to play with a lot in TF2 and sometimes switched to L4D), some of the Red Alert games (in LAN mode), Unreal Tournament 2004 (Capture the Flag, Invasion, etc)... and a few others.
I do like multi-player but generally speaking both co-operation and real life friends (or family members) should be involved somewhere in there, otherwise... not so sure, or I just don't play very long (example, Battlefield 2 or other multi-player games that my real life friends never owned or had the time to play much and ended up selling such games).