The second chord is a chromatic passing chord: the bass line is descending (A -> G#) whereas the top line is ascending (A -> B). It doesn't really have a name which describes it properly (CaugMaj7 is a possibility as is Eaug). The third chord is a true C with G in the bass, so again the bass descends (G# -> G) and the top ascends (B -> C).
So, viewing the first three chords, the bass descends in semitones: A G# G, the top ascends A B C, and the inner voices (C and E) stay constant. Viewed as a sequence from Am to C, the middle chord could be seen as a variety of E (E aug) which is the dominant seventh of Am.
Extra material: I am reminded of musicologist Alan Pollack's comment about a similar chord sequence which is the bridge of "All my loving": In "theoretical" terms, such an augmented chord [the second chord in the sequence] is said to not have a root at all, but is rather the incidental byproduct of melodic motion by an inner voice of the harmonic texture; in this case, from C# -» C-natural -» B; what my jazz-trained friend calls a "line cliche." The fact that it is sustained for a full measure, essentially just as long as any other chord in the song, is what particularly draws your attention to it.... In this case, the sequence is C#m ? E, but it's the same idea. The 'genius' of Page (leaving aside the court case) is to add the ascending line on top which mirrors the descending line in the bass.