This is mostly true, but you're confusing because there are two different levels: corporations are registered and regulated at the state level, of which LLC is one category. The IRS then categorizes registered corporations as either S-corps or C-corps for the purpose of federal taxes. There's not a one-to-one mapping between the two levels, especially since there's 50 different states, but usually LLCs implies S corp.

In addition to @akjj's reply, I also want to point out that it's not entirely true to say that S-Corps/LLCs do not pay taxes. The short version is that a C corp pays taxes a corporate level and individual employees pay taxes on their income, where in an S corp the profits of the business must be either dispersed to the shareholders/owners or reinvested in the business, any profits which are dispersed are taxed as normal income rather than capital gains like dividends would be in a C corp, so depending on the relative size, an S-corp effectively pays a higher tax rate than a C-corp, just obfuscated through its owners.