As others have pointed out, the chord A C E♭ G is A half diminished, which is written Aø7. The primary function of the half diminished chord is to serve as the ii chord in a minor ii-V-i progression. Here are some straightforward examples of minor ii-V-i progressions, all of which use a half diminished chord for the ii:
- Aø7—D7—Gmin (a ii-V-i in the key of G minor)
- F#ø7—B7—Emin (a ii-V-I in the key of E minor)
And here's a web page with more reading on that progression, from a guitarist's perspective. These progressions are called "minor ii-V-i"s because they end with a minor root/i chord. Both the "major ii-V-I progression" and the "minor ii-V-i progression" are extremely common in jazz, and you're almost guaranteed to encounter one or both in any bebop song.
There is an interesting tale or legend behind the origin of the half diminished chord, and we can almost guess this story from the chord itself. Let's consider again the Aø7 chord: A C E♭ G. These exact same notes (A C E♭ G) spell out a very common voicing for F9, a dominant seventh chord in the key of F. Supposedly, when a jazz group was playing/tagging a I-IV-ii-V progression such as F7—D7—Gmin—C7, the jazz bassist would reach the I chord (the F7) and play the third note of the F7 chord (an A) instead of the root note (an F). Musically, this sounded great, because now the bassist is playing A-D-G-C, a series of notes that help make up "the cycle of fourths." So now the progression is F7/A—D7—Gmin—C7. But instead of calling this new chord F7/A, we call it Aø7, thus giving Aø7—D7—Gmin—C7. If we isolate those first three chords, we have a minor ii-V-i progression: Aø7—D7—Gmin. So the half-diminished chord serves as the ii chord in a minor ii-V-I progression, and allegedly it arose from jazz bassists who played the third of the I chord instead of playing the root during a I-VI-ii-V progression.
Given how similar major ii-V-I progressions are to minor ii-V-i progressions, it might be worth showing a few major ii-V-I progressions:
- Amin—D7—GMaj (a ii-V-I in the key of G major)
- F#min—B7—EMaj (a ii-V-I in the key of E major)
You'll notice two main differences: (a) major ii-V-I progressions use a minor ii chord instead of a half diminished ii chord, and (b) major ii-V-I progressions end on a major I chord instead of ending on a minor i chord.