The Quran names the forbidden categories directly: carrion, blood, the flesh of swine, and anything dedicated to a name other than God's at the time of slaughter. Later jurisprudence extends the list to alcohol and to any intoxicant, reasoned by analogy from the Quran's prohibition of wine, and to predatory animals and birds of prey in most legal schools, on the strength of prophetic reports rather than the Quran itself.
Everything not named as forbidden is treated as permitted by default, which is the sense of halal itself: the term marks the default state of food, and haram marks the narrow set of exceptions cut out of it. This is why Islamic dietary law reads as a shorter list of prohibitions rather than a long list of permissions, unlike the more exhaustively enumerated categories of kashrut in Jewish law, which the Quran itself treats as a stricter code imposed on the Israelites rather than a model binding on Muslims.
Necessity narrows the prohibitions further: classical jurists agree that a forbidden food may be eaten to avoid starvation, on the reasoning that preserving life outweighs the general rule. The same logic underlies most disagreements between schools of law, which concern edge cases such as animals of the sea, rather than the core list itself.