Yes, most of the answers seem to be focusing on personal favorites rather than importance to the sport.
To me, the easiest choice is one that, as far as I know, has yet to be mentioned: Aerobics (1968, and subsequent editions), by Ken Cooper. Bowerman's Jogging (1967, I believe) also had a big impact, but Cooper's book (and subsequent work) revolutionized the whole field of exercise by swinging the emphasis to primarily aerobic activity and its importance to overall health (although his later works made a compelling case for shifting more toward strength and balance training in one's later years). In fact, in some parts of the world, "jogging" was actually called "coopering."
Jim Fixx's books were huge sellers, but to me, they simply caught the running wave that was already building to its crest.
For training, I'd choose one of any of a number of books by or about Arthur Lydiard, who revolutionized training for both competitive and noncompetitive runners, and who inspired Bowerman's Jogging. There were some earlier influential books, perhaps by Cerutty and some of the earlier German expositors of interval training, but Cerutty was too quirky and interval-based training was largely pushed aside by Lydiard's base-building and periodization.
Among later training volumes, Martin & Coe, Daniels' Running Formula, Pfitzinger and Douglas, and others were of some use, but I kind of wish that I had never heard of them, because they sought to put training in a rigid four-box system (for Martin & Coe, they were (1) aerobic conditioning (easy, according to others), (2) anaerobic conditioning (threshold), (3) aerobic capacity (VO2max), and (4) anaerobic capacity (repetitions)). In fact, the first edition of Daniels' Running Formula referred to all training between "easy" and "threshold" as "quality junk" running, although he carved out a slightly illogical exception for marathon-pace training. (In his defense, he emphasized that "marathon pace" training was primarily for marathoners, who could justify such training for its specificity neuromuscular coordination and economy at race pace.)
Earlier works (from the 1960s into the late '80s) are worth noting: David Costill's accessible exercise physiology books, Tom Osler's early books, Daniels's early works, George Sheehan's books (I was surprised to discover how many have loved "Running and Being"), Ron Daw's books (the underground classic "The Self-Made Olympian" in particular, but he wrote some fine and insightful work later). Noakes's "The Lore of Running" may be important, but much of it (depending on specific editions) is horribly misinformed. Sandrock's "Running With the Legends" is, I think, very good, but I don't know whether it has been "important." Books by or about Bannister have probably been important in Great Britain, which is certainly significant, but I think they're largely junk and unimportant elsewhere.
Favorite books is another subject. For me, they start with Kenny Moore's "Best Efforts."
This post was edited 4 minutes after it was posted.
I have seen it mentioned but Sub 4:00 Alan Webb and the quest for the fastest mile is a really good book. Gives a similar vibe to running with the buffaloes. I felt it really gave a good insight to his personal journey at Michigan (though short lasting) and how much he truly struggled throughout this time.
Win at all costs is also a pretty good book. Talking basically about the Galen Rupp project as they coined the NOP in the book, its pretty in depth about the history and lore behind the NOP and the eventual disbanding.
I am basing these on current utility and historical impact. A lydiard book isn't in here because lydiard's wisdom can be garnered from letsrun and from websites (famously the "lydiard pdf"). Cerutty isn't in here because of limited modern utility, etc.
There are few legitimate training books I haven't read, at least if they were in print in my lifetime. I've also read many of the classics that are quite difficult to find these days. And a fair amount in adjacent fields (other endurance sports, sprinting, strength and conditioning). There is no book out there that comes remotely close to Magness in terms of summarizing the relevant research as of the date of publication, whatever weird personal beef you have with the guy.
100% on Once a Runner. It’s a book where non-runners would never have interest in, and probably not runners that took up running later in life. But if you ran competitively in high school and college, the nostalgia is strong with this one. The scene where he runs in the rain to his ex gfs sorority house and she thinks he’s crazy…such a great mixture of running obsession and early relationships.