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"Non-immigrant visa" simply means "getting it doesn't grant you immigrant status". In this way it's like a tourist or student visa.

The H-1B is a dual-intent visa. This means "you may apply for it and show intent to immigrate, for example, by starting the PERM process". This is unlike a tourist or student visa, where there are grounds for denial if you show any intent to immigrate.

There are a number of "I'd like to immigrate to the US" visas, primarily based around 3 categories - family, employment, or refugee status. The employment-based green card (which is what H-1B holders apply for) has quotas based upon the applicant's country of birth. An H-1B holder born anywhere other than India or China can get an employment-based green card in less than 3 years.