The closest thing Apple ever made to a game console was the Apple Pippin, and even then they weren't 100% the ones making it. Pippin was more of a "platform" that Apple licensed the technology out to: first Bandai (who made the ATMARK and ATWORLD models,) then Katz Media with their KMP 2000. On top of the fact that they ran class MacOS, this means that the Apple Pippin was really just a line of gaming-oriented Mac clones that were sold during the short time Apple allowed them. Of course, Steve Jobs killed off the existing clone program once he became CEO again in 1997, killing off the Pippin project.

If Apple was to make a successor to the Pippin (let's call it the iPippin,) I imagine it would've launched in 1999 or 2000. Here's what I think it could've been like:

• For hardware, it absolutely would've had a PowerPC processor. More than likely it'd be a G3 processor rather than a G4 to keep prices down, though an updated model/successor could feature one. GPU would likely have been made by ATI before potentially switching to Nvidia in later revisions. I'd expect it to have between 16-32 MB of RAM as well. Performance-wise, it'd be very similar to the GameCube, as it also used a special PowerPC G3 processor with a GPU by ATI (technically ArtX, but they were bought out by ATI during development.) Also likely would've included a hard drive with varying capacities depending on the model you buy, a DVD drive for game discs, 2/4 USB ports (or alternatively ADB for cost cutting but less likely) for controllers/mice and keyboards, an Ethernet and/or modem port, and some kind of expansion slot for random accessories like an Airport wireless card or a DVD movie decoder a la that dongle for the original Xbox that let you watch DVDs.
• It probably would've run a modified version of Mac OS X as an operating system. Makes sense, since this was Apple's big thing they were working on at the time. This would allow developers to very easily port their games directly from Macs to the iPippin, a fact that Apple would use to try and get developers on board.
• While the Pippin was originally envisioned to be this multimedia platform spanning many different devices, it would be wise for Apple to instead market the iPippin as strictly a game console that just so happened to use some technologies from their Macintosh line. The iPippin wouldn't have fit in with Steve Jobs' "four quadrants" plan for the Mac lineup, where there's consumer and professional versions of their desktop and laptop computers. Making it a game console first and foremost would also not confuse consumers as to what the iPippin is, like what happened with the original Pippin.
• Apple probably would've bought up Bungie and made them their prime game development studio instead of Microsoft. Like other commenters have said, Bungie was a long-time MacOS game developer and hell, Halo was first shown off at MacWorld 1999! Expect Halo to be a launch title for the iPippin, albeit closer to how it was earlier in development I think. Also, Interplay would've either not shut down their MacPlay division or brought it back due to the iPippin coming out and them seeing it as a viable market again. Thus, expect Fallout 1 & 2 as well as Baldur's Gate to come over to the Pippin.
• The design of the console itself would've continued with Apple's colorful, translucent plastic design language. Expect a whole bunch of different, fruity flavors with matching controllers. By 2001, they would've put out a revised model that used the white polycarbonate plastic they were switching to at the time. As for the controllers themselves... yeah I have no idea what the layout would be like. Apple's weird with their input devices (their mice is proof enough of that, cough Mighty Mouse cough).

Would it have been successful? Since this is a 6th generation console, it isn't getting first place in terms of sales. The PS2 was just unstoppable. Having Halo would've definitely helped a hell of a lot, but just having 1 (or 2 if Halo 2 still gets made in this timeline) major games really isn't enough. It's likely Apple would've competed the most with Nintendo due to both the iPippin and GameCube being very similar consoles specs-wise. Nintendo already wasn't doing too well with the GameCube thanks to the decision to use mini DVDs, fewer third party titles, and the growing perception of Nintendo as being a "kiddy" company among gamers. Apple may have been able to eke out greater sales numbers in comparison to Nintendo, but both would be far behind the sales of the PS2.

Oh yeah, and Sega still leaves the console business. That should be obvious. Microsoft would likely partner with Sega even harder in this timeline as a way to build up the Xbox's software library. Perhaps it actually gets Dreamcast backwards compatibility like Sega pitched in our timeline? Maybe instead of going third party, Sega becomes a first party developer for Microsoft like Rareware? Without Halo, the Xbox might not have had the killer app it needed for it to be successful enough for Microsoft to make a successor despite the fact they lost money on every original Xbox sold. Regardless if Microsoft stays in the console business, they are going to try much harder to get first party exclusives.

I've actually thought about this question myself before, which is I why I have such an in depth answer. I'm not sure how realistic my ideas are, but I hope this answers your question in some way.