(topo's answer is very good, but I wanted to offer a few other thoughts.)
To understand what "tonal music" is, we have to first define what we mean by "tonality."
The go-to scholarly source on defining it is Brian Hyer's article "Tonality" in The Cambridge History of Western Music Theory. Towards the beginning of his article he states the main difficulty (as I see it):
A further complication (and recurrent tension) has to do with whether the term refers to the objective properties of the music – its fixed, internal structure – or the cognitive experience of listeners, whether tonality is inherent in the music or constitutes what one recent author describes as "a form of consciousness."
But I think he settles on a very good definition shortly thereafter:
the systematic organization of pitch phenomena in both Western and non-Western music.
What's important here is the phrase systematic organization. By this I think Hyer means two things:
First, that the music is hierarchical, and one pitch (the tonic) is more important than all others.
And that this pitch is the focal point on which the music comes to rest.
Hyer might mean a third point, which is the notion of monotonality. If we're privileging one pitch as tonic, then that tonic should be the tonic of the entire piece (hence monotonality, meaning "only one tonic"). This doesn't mean the piece can't modulate, but that it will (with some exceptions) begin and end in the same key.
This means that the definition can be very broad; we can use cadences, but we don't need to; we can use tertian harmony, but we need not; we can use scales, but we certainly don't have to; we can write in major or minor, or octatonic or hexatonic, or something completely different altogether; and so on. What's important is the role of a single tonic pitch that is hierarchically more important than all of the others. There may be different types of tonality (and this different types of tonal music), but it's still tonality; Mozart is definitely different from Wagner, who is very different from Debussy, but they still wrote tonal music. (Note: some may disagree with this paragraph!)
And here maybe it's helpful to discuss how some have defined different types (or "uses") of tonality. Robert Bailey, when studying Wagner, discusses four types:
- Classical tonality, which is that defined by the tonic-dominant polarity. (I return to this at the end.)
- Expressive tonality, which is when chords move up and down by second. Some call this "truck-driver tonality," because the chord switches sound like a revving engine.
- Associative tonality, where particular tonal cues are used as extra-musical references that might not be related to the current stretch of tonality.
- And directional tonality, where a piece begins and ends in different keys. (Contrast this with monotonality, above.)
But even with that knowledge of tonality, there are still disagreements on what is and is not "tonal music." Some, for instance, separate "tonal" music from "modal" music, while others still view modal music as being tonal.
But it may be important to note that two historical theorists, Fétis and Choron, announced "the beginning of tonality" at the appearance of a dominant-seventh chord in a piece by Monteverdi around 1590. Many modern theorists insist on the presence of the dominant in order to define tonality; indeed, they believe that the tonic-dominant polarity is the defining characteristic of tonal music. This mindset largely comes from the work of Austrian music theorist Heinrich Schenker. His influence on our understanding of "tonality" is so vast that college courses on his theories are often simply called something like "Analyzing Tonal Music."