My first computer was an Exidy Sorcerer, bought in the late '70s. It featured a full keyboard including lower case, on screen text of 64 B&W characters by 30 rows, 2 MHz z-80 processor, up to 48 kbytes of RAM, 8 kbyte ROMpacs (an msBASIC pac came standard), ROM-based diagnostic monitor program, etc. As far as full-featured computer, it was significantly ahead of its competitors such as the Apple II, TRS-80, and Commodore Pet. (Prologue because apparently no one here has talked about this computer before.)
It did not have conventional graphics; the computer was always in text mode. A text ROM defined a full set of 256 8x8 characters. However, the top 128 characters were loaded into RAM at bootup. This meant programs could dynamically redefine those characters, giving the computer a limited form of pixel-mapped graphics at a 512x240 resolution. Even within those limitations, clever programmers achieved some very nice effects and even images.
/edit add 2017.01.04 07:35 EST
General comment to many of the answers: I should have excluded the case of a computer having both definable characters and a traditional bit-mapped screen. In that event the definable characters simply allow for non-Latin character sets, and programmers are not forced to jump through hoops to display graphics at the best resolution possible. But when your display is limited only to text mode, programmers need to be pretty darn creative. Especially on the Sorcerer, where the 128 definable characters would only fill in 2 of the 30 display lines in a brute-force approach.
Similarly, computers that had no text mode, only bit-mapped, aren't the same as the Sorcerer either. Of course the characters were redefinable -- everything was! 🙃 This is not meant to downplay the existing answers, which responded to the question as asked. But for any future answers, please limit them to computers with redefinable characters in a text-mode only environment.
Were there other retro-computers that used similar character-defined graphics technology exclusively?