What is a tonal language?
One in which pitch changes the meaning of a word, not just its emphasis or emotion — the same syllable said with a rising or falling pitch is a different word entirely.
How common it is. Far more common than English speakers assume: a large proportion of the world's languages are tonal, including most Chinese languages, Vietnamese, Thai, Burmese, and a great many languages across sub-Saharan Africa and the Americas. Non-tonal languages are arguably the unusual case.
How tone systems differ:
Register tone systems use level pitches — high, mid, low — as in many African languages.
Contour tone systems use pitch movements — rising, falling, dipping — as in Mandarin, which has four tones plus a neutral one, or Cantonese, which has six.
Tone can carry grammar, not only vocabulary. In some languages, tense or possession is marked by tone alone, with the consonants and vowels unchanged.
Why English speakers find it difficult. English uses pitch at the sentence level — rising for questions, falling for statements, stress for emphasis. Learners therefore keep applying pitch for emotion and emphasis, which overrides the lexical tone and makes speech unintelligible. The difficulty is not hearing the tones but suppressing an ingrained habit.
What English speakers do have. Stress placement changes meaning in English too — REcord and reCORD — which is a related phenomenon, and a useful bridge for understanding that suprasegmental features can be lexical.
Related things worth distinguishing: pitch accent systems, such as Japanese and Swedish, where pitch matters but less pervasively than in full tone systems; and intonation, which every language has and which is not tone.
Two interesting consequences. Singing in tonal languages creates tension between melody and lexical tone, handled differently by different musical traditions. And whistled languages exist in several tonal communities, conveying speech over long distances by reproducing tone and rhythm alone.
Speakers of tonal languages show higher rates of absolute pitch, though the causal claim is debated.