I look for different things in different types of games.
I like racers like Wipeoutxl for the psx because you get the feeling you realy are going fast, and the controls are real responsive.
The classic arcade games are great for that fast moving repetative simple gameplay when I dont feel like reading a manual or spending 4 hours getting to the next level of another type of games. Examples are pacman, galaga, space invaders, battlezone, defender, etc. Just dont make them too complicated with too many buttons to have to worry about.
RTS games apeal to me because of the varied strategies you can do to win. To me this is a real planned out type of thinking games where tactics and strategy realy matter. I guess I like collecting resources and building different units. Tech trees are real cool in games, as long as what you choose really affects how the game plays later, too many choices that dont matter just make the game tedious. I also like games like master of orion because of the exploration and conquest involved in them (same as with rts). Dune 2, Age of empires series, first 2 warcrafts, and starcraft. I didnt like warcraft 3 because the defensive structures sucked (basically no defense unless you had an army there), and hero units that killed entire armies basically made all the other units worthless so why bother.
FPS that have nice maps and good but not overpowering weapons are also nice. Capture the flag mode on the facing worlds map in the original UT has got to be one of the best of these type of games. One of the cool things about doom and doom2 was the music and that fact you could have a 100 monsters behind a closed door waiting to kill you plus the gameplay was fast. Its very hard to get the same type of oh shit feeling with the newer 3d games because the computer just cant push the polygons fast enough to have that many monsters going on the screen at the same time at a good framerate (its the hardware dictating the game design).
Im not sure where diablo I/II fit in but mindless hacking and slashing in dungeaons where you get rewarded with magical weapons always kept my attention. There are lots of clones of this type of game that never got my attention, not sure where they went wrong. Note this is a 2d games, the ones that are 3d just dont look right to me.
One of my all time favorite wargames is second front (remake is war in russia) because of the vast playing map, production levels and type of units produced, and the vast possibilities of fighting a huge war in russia between the germans and russians. The graphics are terrible but the gameplay is so cool. Nothing like looking at city objectives knowing if you capture a city with an oilfield your supply of oil will go up, or bombing a city in the hopes you slow down t-34 tank production.
So basically in a nutshell I like games that are extremely simplistic and take 2 minutes to know how to play or deep thinking strategy games with tech trees that make a difference and exploration and conquest being important.