If you assume the axiom of choice, then every cardinal can be well-ordered, and thus can be identified with a particular ordinal (the smallest one of the appropriate size). Because the class of ordinals is well-ordered, it follows that the class of cardinals is well-ordered. Therefore, there is a smallest cardinal greater than any particular cardinal (in particular, there is a smallest cardinal greater than aleph_0).
Without the axiom of choice, the cardinals needn’t even be totally ordered (indeed, this is equivalent to the axiom of choice), but the aleph series is still defined in terms of just the well-orderable cardinals (i.e., aleph_one would still be the smallest well-orderable cardinal greater than aleph_null).
I suppose “smallest one of the appropriate size” is potentially confusing wording; what I mean is “the ordinally-smallest one of the appropriate cardinal-size”.
Perhaps more simply, depending on how you like to look at things, instead of identifying the cardinals as particular ordinals, you can identify each cardinal with the whole range of corresponding ordinals. That is, the class of well-orderable cardinals can be thought of as the class of ordinals modulo the (order-respecting) equivalence relation of equicardinality, with the inherited ordering. This, again, establishes that the cardinals are well-ordered.