The Problem

When initially learning a stringed instrument, players are shown basic chords to learn. Because these are learnt at the top of the neck, the fingering is easier because the fret nut replaces your fingers for some strings.
Also, to make it easy to get started, finger patterns are used that are easier to play. These patterns may not be ones that can be used away from the first fret. (We'll see why soon).

Alas the actual 'trick' to playing anywhere on the fretboard, in any octave, and in any key can be concealed by this method.
We'll use the Ukulele as an example. It has 4 strings.

Here is the traditional tab layout for a C or, more specifically, a C Major.
traditional style C major Tab
To make it a bit clearer below I've added a representation of the neck alongside. (To make the diagram more readable, I've not drawn the sharps and flats). You can see the Tuning at the top. In this case its the standard GCEA Ukulele tuning for a right handed player.
Traditional style Tab C major with Fret labels
"Everyone" learns this as a basic chord because it's very common and very simple. You can see by looking at the neck that the fretted note, on the A string, is in fact a C. This would be called the Root node as the chord is a C chord and the note on the A string is a C. The fact that all four strings are strummed is what makes it a chord (designated with the little 'o' above each string). These are referred to as Open strings.

But this hides the obvious problem, which is that the Nut is taking the place of your fingers because the chord is defined at the Neck. This does make it easier for beginners to learn but the unfortunate side effect is that "most" people will stop when they have learnt a handful of memorised common notes. Unless you can see the underlying organisation you'll have to memorise a new chord every time.

So lets add a little clarity with this version of the diagram (keeping the traditional representation there for comparison).
Compare new style of Tab with traditional one
So in this version we have indicated the roots (there are actually two Root C's in this chord) with little white circles. (Look across at the Neck to see they are C's).
We have also shown the inherent finger positions, which the Nut is doing for us in the original diagram. The intervals are indicated at the bottom of the diagram. (The Major chord is defined as having notes at intervals of Root, 3rd, and 5th but more about that later.)

Is it more complicated - it might seem so, but lets compare what happens when we look at the next note in the scale.
compare C with C# and traditional Tab style
Now we can clearly see something that wasn't apparent before. The C# chord is the exact same shape as the C chord but moved down one fret.
Note that the Root notes are the same C# (on the C string and A string). In fact they are always the same for any shape. This immediately shows us something that might be quite surprising. We can use this shape to play any note on the C or A strings in the Major chord.

So if you want to play a chord - and you can identify the fret associated with that note - and the note is on one of the strings with a root dot on it - then you can play that chord with that shape.

Lets stick with this shape for a moment longer and see what this means.
If we move the shape down one more fret - surely it will play a D Major. Yes it does, and so on until we roll over into the next octave - and keep going.
Showing shape progression down the fretboard

So I'm hoping you're considering that this might mean that there are also shapes for the G and E strings, and that if we knew those shapes, then you could also play the Major chord on any note, on any string.

Lets go on to see how that works and, as a side effect, why we learn to play the D using a different fingering than the one above. (Chord Shapes)