The most basic kind of chord is a triad. It is made up of three pitches, usually each a third apart. In the key of C, the "tonic," or fundamental, chord is C-E-G. Because the C is the first, bottom, or lowest-sounding pitch, the chord in this form is called root position. In traditional western music, the tonic chord is notated by the Roman numeral I. From the tonic, or C in this case, the other pitches are a third and a fifth above.
When the second pitch of the root position chord is on the bottom, or is lowest sounding, the chord is said to be in first inversion: E-G-C in the key of C. The different position of the pitches changes the character of the chord. With the E on the bottom, the other pitches are a third and a sixth above. Because of the latter, the tonic chord in first inversion is notated I6.
When the third pitch of that root position triad is on the bottom, or lowest-sounding, the chord is said to be in second inversion: G-C-E in the key of C. This position of the pitches changes the character of the chord even more. With the G on the bottom, the other pitches are a fourth and a sixth above. Because of this, the chord is notated I6/4. (The six is written above the four.)
Sometimes another pitch is added to the triad to create a more colorful sound. Most often this is the pitch a third above the triad, or a seventh above the root. Seventh chords can also be used in inversions, and they have their own notations. In root position, the chord would be spelled C-E-G-B and notated I7.
As with the triads listed above, the seventh chord can be used with the third, fifth, or seventh on the bottom. With the third on the bottom, the chord is spelled E-G-B-C and notated I6/5. With the fifth on the bottom, the chord is spelled G-B-C-E and notated I4/3. With the seventh on the bottom, the chord is spelled B-C-E-G and notated I4/2.
In popular music notation, the root name of the chord is used instead of the roman numeral: C6, C7, C4/3, etc.