Chromatic mediant is the technical name
This is where the chord roots are a third apart and there is one common tone.
So with Fm and Am you have:
F, A flat, C
A, C, E
So the "C" is the common tone, and F and A chord roots are a third apart.
I think part of what makes the great sound is that the two moving voices move by half steps. Half steps are important with many chord resolutions/tendency tones.
Also, notice in that wiki article the example from Mozart K. 475 where the chromatic mediant relationship and the common tone are used to make a common-tone modulation to a distant key. So on a very large scale you could think of the Fm and Am relationship in terms of a key change.
A little add on for Dm and E.
I assume you mean E (major.)
Let's look at those pitches:
Dm = D, F, A
E = E, G#, B
If we put those in order:
D, E, F, G#, A, B
One thing that should jump out is the F to G#. That happens to be an augmented second. To fast forward a bit... an augmented second comes up in the harmonic minor scale. In this case the G# is the leading tone of the A harmonic minor scale. The full scale being
A, B, C, D, E, F, G#, A. Dm and E function as the
V in A minor.
If you are just going back an forth from Dm to E you are sort of hovering around the dominant
V chord. Typically you would move from the domimant to the tonic
Am. You could try going to Am as part of that progression.