inversions?!?!

Yep, but in such a way that the root note of the chord is no longer the bass note. You invert the order of the notes that build the chord. At least that's how I think of it (and I hope I've got it right).

Here's a description:

Torvund's definition of chord inversions

Inverting something means turning it upside down. If you've got the interval:

E
C

the inversion would be

C
E

When applied to chords, you move one note at a time from the bottom to the top:

G C E
E G C
C E G

so you have as many possible inversions as you have chord tones. Anything else, like

C
E
G

would still be the chord, but not an inversion. Any arrangement of chord tones is a voicing. So... all inversions are voicings, but not all voicings are inversions.

Also, a normal chord (on piano, that is) is like this:

1 - 3 - 5 (Lowest to highest)

If you voice it with the 3 in bass, it is called a 1st inversion:

3 - 5 - 1

Voiced with the 5th in bass, it is known as a 2nd inversion:

5 - 1 - 3

4 note chords, like 7ths, can even be in 3rd inversion:

7 - 1 - 3 - 5

Note though, that the order of the notes, other than the bass note, does not matter. On a guitar especially, the note order can be all over the place, unlike piano. All these chord voicings are 1st inversions:

3-1-5, 3-5-1, 3-7-1-5, 3-1-7-5 and so on, as long as the 3 is deepest.

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Another thing, inverted chords are frequently written as slash chords, for example D/F# (read D over F#).

A normal D major chord is D-F#-A. Over an F# bass note, it becomes a D/F# or 1st inversion.

In the same way, the 2nd inversion could be written D/A and the 3rd inversion 7th chord could be written Dmaj7/C# or D/C#.

I take exactly the opposite approach that Helgi does...

Strict inversions are almost never noted for the guitar, unless it's in standard notation, so you've got a lot of choice in which voicing to play. To my ears, the treble note cuts through the strongest, so I work my voicings around where I want the highest chord note to be. If I want the C on top of a C major, I'll play this:

8
8
9
10
10
x

or this:

8
5
5
5
x
x

without really worrying about what the bass note will be... in an ensemble situation, the bass and/or keyboard is likely to play a note under my lowest anyway, so I don't have complete control over the bottom end.

Slash chords do indicate the bass note, but I think they're over-used in guitar notation. Twenty years ago you saw them rarely, and only when it was important for a walking bass line... today you can hardly pick up a piece of music without seeing at least a few. Since most tunes don't have a strong moving bass that's apparent when you're playing a chord progression on the guitar, I think it's an uneccesary complication... but that's just my opinion.

As far as the order of tones over the root being unimportant, there's some truth to that. When you study theory, the inversions initially follow the 'rules' (as I showed the inversions above), but when you get into harmony, inversions are defined by the bass note. The reason for this is that harmony is written in four parts... that can be actual (four different tones) or implied (where two of the four voices happen to be on the same tone during a chord). Up until about 1800, composers followed the rules for the most part, but over the past two centuries, they've departed a lot - the analysis of a composition through the use of inversions isn't as clear as it once was, and today the bass note alone will determine the inversion for analysis.

NoteBoat, do you agree with hbriem on that --
If you voice it with the 3 in bass, it is called a 1st inversion:

3 - 5 - 1

Voiced with the 5th in bass, it is known as a 2nd inversion:

5 - 1 - 3

4 note chords, like 7ths, can even be in 3rd inversion:

7 - 1 - 3 - 5

:?:

from the link Musenfreund have given --
A chord in root position has a 1-3-5 voicing. When starting from the root position, we can move the root up one octave. We the get at chord with the third in the bottom, fifth in the middle and root on top. This is called the first inversion. The first inversion has a 3-5-1 voicing.
does this mean that only 3-5-1 will be (or should be) considered as the first inversion and not any other (in this case 3-1-5)?

Yes, what Helgi and the link said about inversions is accurate, and it's the way it's taught in theory books (including mine).

When Helgi said that any chord with the third in the root is first inversion, he's getting into harmonic analysis. Four part harmony will have four voices; if they play a triad, there will be a doubled note (or a unison, with two of the voices on the same note). Three voices give two possibilities:

3-5-1
3-1-5

And four voices give a few more:

3-5-1-1
3-5-1-3
3-5-1-5
3-1-5-1
3-1-5-3
3-1-5-5
3-3-5-1
3-3-1-5

Since the 'typical' use is 3-5-1, we can say that any chord which has those notes in sequence is a first inversion (which covers the first three of the eight possibilities of four voices); or we can say that a chord which has 3 in the bass and 1 in the soprano is a first inversion (cases 4 and 7 above).

That covers the majority of the possible voicings - 1 of the 2 triads, and 5 of the 8 four-note voicings with 3 in the bass. Since the other voicings are not strikingly different, modern harmonic analysis treats them the same way.

On the guitar, I prefer to think of voicings - the sequence of each voice - rather than inversions. The guitar is not well adapted to close position chords; we tend to spread things out a bit more because of limitations of the instrument. Since we're spreading things out, I find very little difference (to my ear, anyway) between these two voicings of G:

3 3
0 0
0 0
0 0
2 2
3 x