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Alternative is Rock Music made outside the mainstream. They Might Be Giants, Joy Division, Smashing Pumpkins, Radiohead, and Sonic Youth are examples of alternative. It has lots of subgenres, like indie, post-punk, goth, and Britpop.
Metal is a genre focusing on crushing riffs, musical complexity, raucous or shouted vocals, and dark, rebellious lyrics. It has its roots in blues-rock and louder prog. Everything from loud, droning instrumentals to hyper-fast growled drum-guitar showdowns is considered metal. Napalm Death, Metallica, Black Sabbath, and Iron Maiden are metal bands.
Punk is a stripped-down, angry guitar-based genre with themes of politics, drugs, and the punk scene itself. It uses a lot of distortion and is very lo-fi and fast, but usually quite melodic. Types of punk include Celtic punk, hardcore, funk-punk, and psychobilly, which is punk rockabilly. Bands include Black Flag, the Ramones, early Devo, The Fall, the Sex Pistols, Bad Brains, and the Clash.
I'm not sure what alternative-punk-metal would sound like. I'm guessing sort of like Primus, Jane's Addiction, and Rage Against the Machine. Early Nirvana and Soundgarden is alt-punk-metal, as is Alice in Chains, but these bands are known as grunge bands.
