Technically speaking, an augmented chord is a major chord with a raised (sharpened) fifth. So for an augmented triad we would have the scale formula 1- 3 - #5. This equates to the notes C, E, G# for the C augmented chord.
Augmented chords are most often labelled as "aug" or "+" I.e. C aug or C+.
Problems can arise with this chord because there is sometimes a lack of consistency in the way people name them. The term "augmented", when applied to musical intervals, means to raise by one semitone. This means that the terms augmented fourth, augmented fifth, augmented eleventh etc.. they all make sense, they simply mean #4, #5, #11.
So why can this be confusing? Because the term "aug" or "+" when used in conjunction with a triad is taken for granted that it is referring to the fifth. For extended chords, this isn't always the case. Take the augmented seventh chord for example. Is this a dominant seventh chord with a raised fifth or a dominant seventh chord with a raised seventh? The name Caug7 can be rather ambiguous as it suggests neither in any certain terms. If you've been revising your chord theory so far then you have probably already realised that a dominant chord with an augmented seventh would actually be a major seventh chord so this one is rarely open to confusion, even though, you will often find this chord written as C7+5 which takes away any confusion, it is a C7 chord with an augmented fifth.
Another example would be the augmented ninth. Most of the time this chord name will be referring to a seventh chord with an added raised ninth. Even though you will sometimes see it written as Caug9 or similar, the better way to name it is C7+9.
The augmented triad contains notes of an equal distance apart just like the diminished seventh does, but with the intervals a major third apart (four semitones). Each of them can be considered the root, so like the diminished seventh, it's context dictates the key.