RTS, FPS, (MMO)RPGs > Almost always better on PC
Platformers, Sports, Action/Adventure Games, Racing/Flying, Fighting > Almost always better on Consoles
I could exhaustively elaborate on this for eternity, but I will try to sum it up as best as I can.
RTS FPS and RPG games usually requires dozens, some requiring up to hundreds of different button combinations in order to use all available abilities. The only way to make these games work on console is to dumb down the complexity of them and vastly modify their control schemes.
Someone cited Halo Wars as a successful example of Console RTS. I'd say Halo Wars is a perfect example of a dumbed down RTS system vastly modified to fit an unnatural control scheme.
Here's how the control is described on the Halo Wars wikipedia entry:
"Designed specifically for the Xbox 360, Halo Wars uses the console controller's A-button for unit selections. A single tap of the A button selects one unit, a double-tap all units of a like type, and holding down the button creates a paintbrush-style selection cursor. The X-button for moving to or attacking a target, and the Y button activates special abilities.[7] The D-pad is used for navigating to battles or cycling through bases. Buildings and upgrades are arranged and managed in a circle-shaped menu.[8]"
Here's how I would describe the control of a computer RTS:
"Use mouse, highlight all units, click keyboard key or mouse action to do something with them."
FPS games on consoles are usually are dumbed down with auto-aim and limited content. Since it is impossible to mimic the precision of mouse aiming on PC FPS with a console control stick, most games just resort to letting the game aim for you automatically, which takes the majority of the skill and fun out of the game. Additionally, most FPS games on PC thrive on user created content, levels, mods, design packs, additions, and none of that is available to most console titles.
RPG games are a mixed bag, depending on how they are designed you can play them equally on either system, but MMORPGs can only be played on PCs because the complexity of control (and communication) requires too many buttons to ever rationally put on a controller. I would like to see someone try but I don't think it will ever work.
-----
Platformers, Sports, Action/Adventure Games, Racing/Flying, Fighting
These games are almost always better on consoles. Platformers are ideal, control sticks and few buttons are all you need for a good platform jumper. Sports games are great on consoles and most companies rarely even release versions of their Sports games on PCs anymore. General Action/Adventure games are a mixed bag, but more end up on consoles than PC by a wide margin. Racing and Flying are almost always better with a stick/controller than a mouse and keyboard. Fighting games are designed perfectly for consoles, because most games require only a stick and 6-8 buttons.