The cardinality of the first infinite ordinal, omega (the number of natural numbers).
Aleph 1 is the cardinality of the smallest ordinal whose cardinality is greater than aleph 0, and so on up to aleph omega and beyond. These are all kinds of infinity.
The Axiom of Choice (AC) implies that every set can be well-ordered, so every infinite cardinality is an aleph; but in the absence of AC there may be sets that can't be well-ordered (don't posses a bijection with any ordinal) and therefore have cardinality which is not an aleph.
These sets don't in some way sit between two alephs; they just float around in an annoying way, and can't be compared to the alephs at all. No ordinal possesses a surjection onto such a set, but it doesn't surject onto any sufficiently large ordinal either.