If you are an engineering undergraduate or an engineer coming to the subject cold, read An Introduction to Combustion by Stephen Turns. It is the standard mechanical engineering course text, it assumes thermodynamics and a first course in fluid mechanics, and it introduces chemical kinetics from scratch rather than expecting it. Fundamentals of Combustion Processes by Sara McAllister, Jyh-Yuan Chen and Carlos Fernandez-Pello is the shorter alternative: a compact senior-level text, lighter on applications and quicker to get through.
The single most useful thing to know about this subject is that the books sort almost entirely by prerequisites, and the titles do not signal them. Combustion sits at the intersection of thermodynamics, chemical kinetics, mass and heat transport, and fluid mechanics, and each author assumes a different subset. Open the wrong one first and the problem is never the topic — it is that page thirty is doing asymptotic analysis you have not met.
Two books both called Combustion
Combustion by Irvin Glassman, later editions with Richard Yetter, is the graduate text that leans chemical. It is strongest on flame structure, ignition, detonation and the chemistry of pollutant formation, and it expects you to be comfortable with thermodynamics and reaction kinetics; the fluid mechanics is present but not the emphasis.
Combustion by Jürgen Warnatz, with Ulrich Maas and Robert Dibble, is the other one, and it leans toward detailed chemical mechanisms and numerical simulation. It assumes transport phenomena and kinetics and is the more natural bridge for a reader heading toward computational work. Neither substitutes for the other; if you can only take one, choose by whether your next step is experimental chemistry or simulation.
Combustion engineering by Kenneth Ragland and Kenneth Bryden is the practical counterpart to both — furnaces, boilers, solid fuels, engines, emissions — and is the right book if your interest is equipment rather than theory.
The graduate theory shelf
Principles of combustion by Kenneth Kuo is very large, mathematically heavy and organised as a reference; it assumes fluid mechanics and transport at graduate level and is more often consulted than read. Combustion theory by Forman Williams is the classic theoretical treatment and the one specialists cite most; it is built on activation-energy asymptotics and applied mathematics, and it is not a first book under any circumstances.
Combustion Physics by Chung Law is the modern unified graduate treatment covering the same territory with more recent material and a somewhat more systematic build-up. Combustion fundamentals by Roger Strehlow is an older graduate text, still clear on flame propagation and detonation, though superseded on chemistry.
Specialised research territory
Once the fundamentals are in place the subject splits by phenomenon. Turbulent Combustion by Norbert Peters is the concise research monograph on turbulent flame regimes and the flamelet approach, and it assumes turbulence theory. Theoretical and numerical combustion by Thierry Poinsot and Denis Veynante is the standard reference for large-eddy and direct numerical simulation of reacting flows — the practical book for computational combustion.
Chemically Reacting Flow by Robert Kee, Michael Coltrin and Peter Glarborg connects the governing equations to the numerical machinery of chemical kinetics software, and is the book to have open beside a simulation. Detonation by Wildon Fickett and William Davis is the specialist treatment of detonation waves, mathematical and narrow. Gas turbine combustion by Arthur Lefebvre is a practitioner design reference for combustor engineering rather than a course text, and it is the one book here you would find on a working engineer's desk.
A sensible route
Turns or McAllister for foundations. Glassman or Warnatz for the graduate treatment, chosen by direction. Williams or Law when you need the theory properly. Then Peters, Poinsot, Kee, Fickett or Lefebvre depending on whether your problem is turbulence, simulation, kinetics, detonation or hardware. Skipping the first tier and starting at Williams is the most common way people conclude that combustion is impenetrable, and it is not.
Follow the full path and each book arrives after the one that makes it readable.
Follow the full ordered path here: Best Books to Learn Combustion, from Undergraduate Text to Research Monograph.