Dm chord inversions
Dm/F and Dm/A are the first and second inversions of the D minor. This means that the bass tone is shifting from D to F or A.
Try in a chord progression
Gm - Dm/F - Eb
Dm/A - Bb - C
When stumble upon these chord names, you can choose to ignore the alternate bass note and play the regular D minor. It will always sound ok, but it may lack some subtle additions to the sound. Notice that the Dm/A and the regular Dm are identical except that the open A-string is included in the former.
Apart from the diagram version, two other options for Dm/F are 1X023X, which can be preferred for fingerstyle, and 100231.
Theory of the Dm/F and Dm/A chords
Both chords consist of the notes D, F, A. The difference is that the order of notes shifts. F is the bass note in the first inversion and A is the bass note in the second inversion.
Alternative chord names
Dm/F is identical with F6(no5).
See also Dm7 inversions
Chord constructionDm x - x - D - A - D - F
Dm/F x - x - F - A - D - F
Dm/A x - A - D - A - D - F
Guitar versions of the chord
Notes in chordDm D - F - A
Dm/F F - A - D
Dm/A A - D - F
Note structure in theory