It is trite but these are the three well known chords. Who had time to learn more than three songs already know that there are really not three but five chords! And this is the truth but very simplified… Really these first chords Am|Am|Dm|Dm|E|E|Am|Am:|| are a part of the harmonious diatonic guitar structure. Five chord forms are triads.
Five major shapes correspond to five minor shapes. Three of those (E shape, D shape, A shape) are very propitious for the transformation of a major chord into the minor one of the same name. Another two (C shape, G shape) can be transformer into the parallel minor only partially. This circumstance is closely related to the shape's and triad's construction. Pay attention to the structure of the E chord in the Major-Locrian position.
The note E is on the first string,
the note B is on the second string,
the note G# is on the third string,
the note E is on the fourth string,
the note B is on the fifth string,
the note E is on the sixth string.
So a note E is on the first, third and the sixth strings, a note B is on the second and the fourth strings, a note G# is on the third string. This shape contains three tonics, two fifths and one third. If we analyze similarly all chord shapes, we will see these consist of three notes (I, III, V - mode degrees), these are triad inversions. The Dorian position, the D shape contains two tonics, two fifths and one third. The Phrygian-Lydian position, the C shape contains two tonics, one fifth and three thirds. The Mixolydian position, the A shape consists of two tonics, three fifths and one third. And the Natural Minor position, the G shape contains three tonics, one fifth and two thirds. Shapes which have in their structure one third (E, D and A shapes) convert into the parallel Minor by alteration of one note - the third degree. Forms G and C (which have in their structure correspondingly the doubled and tripled third degree) have more difficulties with the transformation into the parallel Minor.
When we say a triad, we mean firstly that a chord consists of three notes, but the number of sounds can be more than three. We can double or treble these notes, change the arrangement inside of a chord (a triad inversion), these operations don't influence on the harmonic function of a chord and its name remains the same.
There are only five different chord shapes on the guitar. Any chord you can only imagine will be one of these five shapes. The below picture shows all five chord shapes in the major and minor variants (ten fingerings altogether). Chords shown in the first position on the guitar are called correspondingly to chords they express in this position. If we "relatively" replace the guitar nut by using the barre, these chord shapes can be used across the neck. It depends on the bass, if we know the note from which the fingering is built, we can use it.