What are chords and chord inversions?

General definition of a chord

A chord is the layering of several tones played simultaneously - usually built on superposed thirds. Chords are defined by their root note and their quality (major, minor, 7, etc) - and eventually by their inversion.

Triads

A triad is a chord made of three notes. It consists of a root and the third and fifth above it. The most common triads are:

Major - has a major third and a perfect fifth
Minor - has a minor third and a perfect fifth
Diminished - has a minor third and a diminished fifth
Augmented - has a major third and an augmented fifth

On the staff below, the C major, C minor, C diminished and C augmented are transcribed:

Seventh chords

These chords are triads with a supplementary diatonic third superposed. The most common seventh chords are:

Dominant 7 (or just "7") - is a major triad with a minor 7 added.
Minor 7 - is a minor triad with a minor 7 added.
Diminished 7 - is a diminished triad with a diminished 7 added.
Half diminished - is a diminished triad with a minor 7 added.

There are more "triad + 7" combinations than mentioned here. You can find them in the chord definition table of EarMaster's Activity Setup. 

Chord inversions

The regular position of a chord with its root note at the bottom is called the root position.
When the third of the chord (e.g. E in a C major triad) is at the bottom, it is said to be in first inversion. When the fifth of the chord (G in a C major triad) is at the bottom, it is said to be in second inversion.

On the staff below, the C major chord is shown in root position, 1st inversion and 2nd inversion: