A standard major chord can be composed of the root of the major scale, the third of the major scale, and the fifth of the major scale. In the key of A, for example, those notes are A, C#, and E. If, instead of the fifth of the major scale you take the augmented fifth – the one that is half-step higher than the fifth, you will get an "augmented chord" (also known as an "augmented triad"). The augmented A chord, for example, is A, C#, F.
An augmented chord, also known as an "augmented triad", is composed of three notes where there are two full steps (four semitones) between the first and the second note and two full steps between the second and the third note.
Another way to say this is to say that the chord is composed of a major third (the two full steps between A and C# in the above example) and another major third (the two full steps between C# and F in the example). Alternatively as above, you say that the chord is composed of a root, a major third, and an augmented fifth.
The chord uses many notations, such as A5 or A#5, Aaug, or A+.
The augmented chord is considered dissonant or atonal as it lacks a tonal center or a root. For example, the augmented Aaug chord as above is composed of A, C#, F, the augmented C#aug chord is composed of C#, F, A, and the augmented Faug chord is composed of F, A, C#. Thus, the three augmented chords Aaug, C#aug, and Faug are all composed of the same notes. It is difficult to figure out what the root of the chord is, what scale it belongs to, and what tonal center or root it drives to and hence the chord is dissonant.
The augmented chord does not occur naturally often as the most common scales (the major, the natural minor, etc.) do not contain three notes the steps between which are as described above. One example where the augmented chord does occur naturally is the melodic ascending minor scale. The melodic ascending minor in the key of A is A, B, C, D, E, F#, G#, and in this scale the augmented chord is Caug composed of C, E, and G#.