At the beginning of the first measure there is a chord with the notes of Eb, G, C#, and G.
What is this chord?
I suppose that the song is in B minor and I can't find any chords even in Modal Interchange chords that fits that chord in my knowledge.
It also appears in measure 3 with an inverted form.
It keeps on appearing on the song and I can't recognize what chord it is.
To fully understand this chord, I think it's best to look at this piece in the context of the entire work.
More specifically, this is the second movement, and it's in B minor. But it's preceded by a movement in G major. As such, I'm of the opinion that this opening is Tchaikovsky transitioning from the G major of the prior movement to the B minor of this movement.
As such, I think the first two chords are in some sense still in G major; we just heard a movement in G major, so our ears will hear these first two chords as still being in G. They are thus a Ger+6 moving to V.
Measure 2 is a vii°7 that leads to D major. But it's important to note that this chord is also the vii°7 that leads to B minor, which is our ultimate goal. In fact, this chord returns, re-spelled and inverted, in m. to make our first clean movement to B minor.
As for the chord on the downbeat of m. 3, I view this as an inversion of the Ger+6 from the first measure. It's an odd inversion, but I think there's a larger pattern here:
Notice that the second beats of the first three measures are just different inversions of D major. The first beats of these measures are chords that lead into D major. As such, these first three measure (and this will continue later into the piece) just prolong D major as a transitional harmony between the prior movement's G major and this movement's B minor.
Just looking at the notes, this could be an augmented 6th chord (my suspicions were aroused when I saw the C♯).
Augmented 6th chords are built from a triad with the augmented 6th interval, so E♭-G-B♭-C♯ (note: this is enharmonic to an E♭7 chord, but it's written with the A6 instead of the m7 interval). This describes the German Augmented 6th. There's another type called the Italian Augmented 6th, which is just like the German but lacks the fifth (E♭-G-C♯) That sounds pretty close to what you just described.
Possibly the French augmented 6th, which has E♭-G-A-C♯, changing the 5th to an augmented 4th. In your example, that A appears in the next bar, so maybe that's it.
The case against this being a +6 chord would be that this doesn't resolve like one. Usually, these chords would be built on the ♭VI and resolve to the V chord. However, this is built on the ♭II and resolves to the I. Similar, and perhaps this is a temporary tonicization of the IV chord, G major, but possibly not. It could be simply a new use for the +6 chord, but that brings into issue spelling.
So far, I've only been considering that the notes are all spelled correctly in the sheet music. If they're not, then my best guess is either a tritone substitution for the dominant or a Neapolitan 6th chord leading back to D. That, or A7♯11. A lot of the analysis will depend on genre, evidently. (Actually, that A7♯11 would be spelled correctly, so if this were jazz, this would almost certainly be correct.)
Side note: The different types of Neapolitan 6th chords can be chalked up to voice-leading. In C major, Gr+6 is A♭-C-E♭-F♯, which resolves to G:
The It+6 is the same, but getting rid of the E♭ to D (no parallel 5ths!). Usually, the E♭ turns into another C which resolves up to D.
The Fr+6 replaces the E♭ with a D, resolving it to D, obviously. This also solves that parallel 5ths violation that the Gr+6 has (though I believe it was used anyway, composers treated it as an exception to that rule).
To sum up, the Gr+6 resolves its 5th down a half step, the It+6 resolves it up a whole step, and the French leaves it alone to get to the chordal fifth on the dominant chord.
