Abaug7 is Ab, C, E and Gb. Start with and raise the 5th from to . You’ll also see this chord written Ab7♯5 or Ab7(♯5).
Hear that one-note change by comparing with Abaug7.
Only the 5th changes: E♭ rises a half step to E. The other notes stay in place.
When Abaug7 resolves to , that can rise another half step to .
Also written Ab7#5, Ab+7 or Ab7(#5).

Abaug7 in 1st inversion puts C in the bass instead of Ab, the 3rd of the chord. The notes are the same; what changes is what the bass line does, which is most of why you would reach for it — it lets the lowest voice move by step where the root would have jumped.
One of Abaug7’s 14 shapes does this:x-x-10-11-9-12
Abaug7 is a dominant with its tension raised, so it still points at — harder than a plain seventh does. A plain dominant has two voices a semitone from home; an altered one has four, because every altered tone is also sitting a semitone from a note of Db. More voices leaning the same way is all "more tension" means here.
Contains Abaug7 and adds ninth and flat fifth.
Contains Abaug7 and adds flat ninth, sharp ninth and flat fifth.
The common shapes and the derived voicings come from different places, and both are explained in how this site is built.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.