Banda is a kind of band that’s all brass plus a clarinet and / or saxophone. At the bottom is a tambora (a low tuba) and a regular tuba, then a small baritone horn or two, trombones, trumpets, and the clarinet or saxophone.
Mariachi has at least 2 violins, two trumpets, a Spanish guitar, a vihuela (high-pitched 5 string), and a guitarón.
Norteño consists of at least a bajo sexto (12 string guitar) and an accordion.
Where the confusion may arise is that all of these bands play the same kinds of music on different instruments. These types include the ranchera, the cumbia, the polka, the corrido, and others.
I don’t know much more, as I’m not really a huge fan of traditional Mexican music (and I’m also confined to my PDA this week, so typing ain’t easy), but hopefully someone else may come along to add some samples or something.