Quick HDD primer for upgraders
PS4: Up to 2TB+, internal 2.5" drives only (9.5mm or thinner)
Recommended: Seagate Backup Plus Slim 2TB
Sony lets you upgrade the internal drive easily, without killing your warranty, but doesn't allow external drives to install games. The Backup Plus Slim may be an external drive, but there are easy youtube videos out there detailing how to crack it out of its case (easy, I did it) and get it working on your PS4. It's also really cheap, as low as $80-$90 in seasonal sales, and inexplicably cheaper than the internal laptop version you'd see on NCIX/Newegg.
PS3: Up to 1TB, internal 2.5" drives only (9.5mm or thinner)
Recommended: Seagate Backup Plus Slim 1TB
Same idea. Spinpoint was what used to be recommended for PS3, and that division was bought by Seagate and put into these very drives. (I've heard 1.5TB also works for PS3, and some people have 2TB working, but 1TB seems to be the "no nonsense" limit right now. YMMV.)
Wii U: Up to 2TB, external only
Recommended: WD MyBook 2TB
Wii U only has flash memory it in (no HDD) so you have to go external here. The USB ports may not supply enough power for any small HDD to work so I really recommend you go with a cheap WD MyBook - these are vaguely book-like black enclosures with 3.5" drives that come with their own AC adapter to supply power.
XBOX ONE: 5TB+, external only
Recommended: WD MyBook 4TB
The One has a special USB port on the side of the system for plugging in external drives, and almost anything will work so long as it's USB3. Since the limit is huge I would recommend a MyBook but really any 4TB external is going to work (and AFAIK every 4TB external today is 3.5", which means a power brick is coming with it...) You could go small, with a 1TB or whatever. There are some specially branded mini drives for XBox One. But going big at this form factor is pretty cheap.