Humanism is sometimes described as an ethical standpoint in which we have a right and obligation to discover the meaning of our own humanity - this being considered the highest form of thought we can achieve. In a more extreme form, humanists may consider human values to be the highest moral values in existence.
Ahumanism would oppose that standpoint. It might include the ideas that our humanity is imponderable, or meaningless, and perhaps inherently ignoble and amoral, or dwarfed by greater values or equally good values found outside human thought.
In this latter respect, many theistic religions may describe themselves - or else be describable - as ahumanistic, depending on the degree to which they hold that humans can be godlike.
But similarly, many secular thinkers consider humans to be far too self-absorbed - even dangerously malign to the natural order. They might hold human existence to be simply an accident in an evolutionary series of accidents. They might consider nature or evolution itself as being greater and more noble than just its human part. Those thoughts too might be considered ahumanistic.