Typically it means “choice” or “parallel” in that a <|> b is either a “choice” of a or b or a and b done in parallel. But let’s back up.
Really, there is no practical meaning to operations in typeclasses like (<*>) or (<|>). These operations are given meaning in two ways: (1) via laws and (2) via instantiations. If we are not talking about a particular instance of Alternative then only (1) is available for intuiting meaning.
So “associative” means that a <|> (b <|> c) is the same as (a <|> b) <|> c. This is useful as it means that we only care about the sequence of things chained together with (<|>), not their “tree structure”.
Other laws include identity with empty. In particular, a <|> empty = empty <|> a = a. In our intuition with “choice” or “parallel” these laws read as “a or (something impossible) must be a” or “a alongside (empty process) is just a”. It indicates that empty is some kind of “failure mode” for an Alternative.
There are other laws with how (<|>)/empty interact with fmap (from Functor) or pure/(<*>) (from Applicative), but perhaps the best way to move forward in understanding the meaning of (<|>) is to examine a very common example of a type which instantiates Alternative: a Parser.
If x :: Parser A and y :: Parser B then (,) <$> x <*> y :: Parser (A, B) parses x and then y in sequence. In contrast, (fmap Left x) <|> (fmap Right y) parses either x or y, beginning with x, to try out both possible parses. In other words, it indicates a branch in your parse tree, a choice, or a parallel parsing universe.