: What do you call a chord that is not complete? What do you call a chord that is not complete or how can you tell what chord a group of notes are? Is it by looking at the root note? For example: C# E G# B F# =>

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I'll address the B F G question first... the simplest way to put these notes into a "common practice" chord is to arrange them with G as the root, i.e. G B (D) F == G7. Try spelling the chord with B or F as the root for comparison! Yuck.

Also, it is quite common, when voicing a dominant 7 chord with only three voices, to eliminate the fifth (D in this case). This is because it isn't critical to the "quality" of the chord and having it doesn't really add anything. A perfect fifth above the root is pretty much implicit. So the fact that there is no D in what we are calling a G7 is not a surprise, in fact I'd say it strengthens the case for it being a G7.

With B in the bass, this is a first-inversion chord.

Finally, what does your ear tell you? Play these chords back-to-back on the piano:

C (Maj) : C E G

then

G7 / B : B F G

Does this sound familiar? It should. It's a simple I-V in C.

For many of the other examples, the answer to "what is this chord" depends a lot on context. What comes before, and probably more importantly, what comes next?

B C# F#

Without context (and to some degree, what's in the bass), it's impossible to say if this is a B Major-type of chord, with an added 9 (C#) for color; or a C# Dominant-type of chord, with the F# hanging over from the previous chord, waiting ("sustaining") to move to E# and form C#7, the dominant chord in F#/f#-; or an F# chord, with the B sustained from the previous chord, waiting to move to A# to form F# Major (or to A to make f#-); or, perhaps, something else entirely!

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By definition all chords are "complete" if they have 3 or more notes performed at the same time. But depending on how it is used (ie is there a pedal, chords/melody coming from and progressing to etc...) will help you understand its function and therefore can help you in naming it.

Function can be thought of as chords that sound at rest or chords that sound tense in the listeners ear.

Another way to think of function is tonic chords (either M or m) and dominant chords (V7 chords). Tonic chords are typically chords that are used for resolution and dominant chords are the chord types that need resolution.

All this was standardized way back by the great master JS Bach himself and we still seem to appreciate music based on this formula. (Just to make things a little more complicated there is another type of chord function and that is the pre-dominant a chord that leads to the dominant).

So you see you can not just ask what is the formula for figuring out a chord (set of 3 or more pitches) it has to thought of in its entirety.

So the better question would be how does this group of notes function?

Hope this helped.

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Not every group of notes has a name, but with most you can put together some name. Your first unknown E, F#, and G# can be looked at an Eadd9 (thanks Charles). Your last example B, C#, and F# is just a Bsus2.
A chord is technically any set of 3 or more notes so technically any group of 3 or more notes is a chord regardless if it seems to be missing notes or not. Most chords are nameable and all set of notes can be reduced to a prime form which is a very computational way of looking at chords.
To name chords based on notes can sometimes be tricky especially since some groups of notes yield multiple chord names, but there are ways to look at groups of notes to see what chords they make. Here are a few things to remember when looking at groups of notes to name chords:

The bass note is not always the root of the chord.
Chords are typically built in 3rds, so try and stack the given notes in thirds.
The 5th of a chord can typically be omitted.

A little trick I use when naming chords is to take the note I think is the root of the chord and compare it against the major scale with a lowered 7th (the Mixolydian mode). The reason I do this is because how common the dominant 7th chord is and how extentions typically line up very well with this scale. I'll show you with the examples givin.
EX1 - C# E G# B F#
We can see that the first 4 notes stack up nicely in 3rds so let's assume C# is the root. Now we compaire all the notes to the C# Mixolydian scale:

C# D# E# F# G# A# B
C# X E F# G# X B

As you can see we have a 3rd, 5th and 7th. The 3rd is lowered which is fine it just means the basic triad is minor. We don't have a 9th or a 13th, but we do an 11th. From this information we can call the chord a C#m11.
EX2 - E F# G#
We only have three notes in this chord and they are all next to each other step wise. We can guess that is E is the root and let's compare the notes to the E Mixolydian scale:

E F# G# A B C# D
E F# G# X X X X

As you can see we have a 3rd, but we are missing a 5th and 7th. Missing the 5th is not a big deal, but because there is no 7th any extensions need be considered "added". There is one extension a 9th so from this the chord is an Eadd9.
Will finish the others soon...

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The first one can be E6/9. The instrument it's played on is inconsequential. More to follow.

EDIT: 5 yrs on! E F♯ G♯ - an E chord with no 5th, common enough. There is a M3, so major of some sort. The F♯ is extra, not suspending the M3, so it's Eadd9.

B F G won't be a B chord - the only diatonic note there would be B! Try G7. It has root - G, M3 - B and ♭7 - F.

B F♯ E♯. If root is B, then E♯ isn't diatonic. Next to F♯ might have made a maj7, but that would need root of F♯. No real useful chord, probably a passing note 'chord'.

B C♯ F♯. Simple Bsus2. B is root, F♯ is P5, leaving C♯ as the M2, but since there's no 3 of any kind, the M2 is called suspended. Often it moves in the next harmony to the 3. Thus Bsus2.

As in my recent comment, the first chord isn't C♯ major anything. With an E, it must be minor if the C♯ is root. E6/9 seems a reasonable name - more exactly E6/9/C♯.