What are drop voicings?
Drop voicings are formed by taking a close voicing and dropping certain notes one octave. A drop-n voicing drops the nth note, counting from the top, one octave.
For example, a drop-2 voicing of a CMaj7 chord can be formed by starting with a close voicing C-E-G-B, i.e. a stack of thirds, and dropping the 2nd note from the top one octave to get G-C-E-B. This is a drop-2 voicing of a CMaj7 in second inversion.
You can find other inversions of the drop-2 voicing by raising each note to the next chord tone:
G-C-E-B --> B-E-G-C (drop-2 CMaj7 in third inversion)
B-E-G-C --> C-G-B-E (drop-2 CMaj7 in root position)
C-G-B-E --> E-B-C-G (drop-2 CMaj7 in first inversion)
It is worth noticing that the drop-2 voicing of the Maj7 chord arranges the notes as 1-5-7-3. You can form other drop voicings the same way. Drop-2, drop-3, and drop-2&4 voicings are probably the most common on the guitar.
How to determine which notes have been dropped in a given voicing
Taking your example E-A-C#-F#, you can rotate through the inversions:
E-A-C#-F# --> F#-C#-E-A (you might already recognize this as a drop-2 with 1-5-b7-b3)
F#-C#-E-A --> A-E-F#-C#
A-E-F#-C# --> C#-F#-A-E
The last inversion has the last three notes arranged so that raising the bottom note one octave (an inverse drop-2 operation) gives F#-A-C#-E, a close-voiced F#min7 chord. This means that E-A-C#-F# is a drop-2 voicing of F#min7 in third inversion.
Similarly, for A#-F#-B-D#:
A#-F#-B-D# --> B-A#-D#-F#
B-A#-D#-F# --> D#-B-F#-A#
The last inversion can be put into a close voicing by raising the D# one octave (an inverse drop-3 operation) to get B-D#-F#-A#. So A#-F#-B-D# is a drop-3 voicing of a BMaj7 chord. Notice the pattern here, a drop-3 voicing in root position is 1-7-3-5.
To figure out if you have a drop voicing, you can either remember the patterns for drop voicings in root position (or some other convenient inversion) and invert an unknown voicing until it matches a known pattern, or invert the unknown voicing until a sequence of inverse drop operations will put the chord into a close voicing.
There are lots of other ways to voice 7th chords; in particular, chord voicings with doubled notes don't fit the usual way of defining drop-voicings. Voicings that drop a note from a close voicing more than one octave are also not treated by this system. Still, drop voicings are a good beginning for developing a vocabulary of chords.
Down the rabbit hole
Before looking at how to apply all of this to the guitar, anyone who is interested should look at the answer by @MichaelCurtis for a great way to visualize these and a great alternative solution to finding out if a particular chord voicing is a drop voicing.
How to apply this on the guitar
Knowing how to spell a drop voicing is one thing, but knowing how to construct it on the fretboard is another. As an example, I'll construct the drop-2 voicings of a CMaj7 chord on the fretboard. The result will be a collection of four voicings (drop-2 in root position and three inversions), some of which may look familiar, and some of which may not.
Begin by constructing a close-voiced CMaj7 (C-E-G-B) on the top four strings; this is the first chord block in the diagram above. Next, drop the second note from the top (the G on the second string) one octave, to the 10th fret of the fifth string. The second chord block in the diagram shows this chord. The fingering is not impossible, but it is a little awkward, so let's adjust by moving the B on the first string to the (now vacant) second string; this is shown in the third chord block. This is a drop-2 voicing of the CMaj7 chord, and since G is in the bass it is a CMaj7 in second inversion. The notes of this chord are G-C-E-B.
To find the inversions, move each of the notes along the string that they are played on to the nearest chord tone. The third inversion is found by moving the voices up one chord tone (first chord block below): G-C-E-B --> B-E-G-C. The first inversion is found by moving the voices down one chord tone (second chord block below): G-C-E-B --> E-B-C-G. As an aside, I am very fond of this voicing of a Maj7 chord because of the minor 2nd in the inner voices. The root position is found by moving the voices of the first inversion voicing down one chord tone (third chord block below): E-B-C-G --> C-G-B-E.
You can't construct these every time you need them, so learn the shapes and learn where the root, third, fifth, and seventh are in each of the inversions. Learn how these shapes change when played on the top four strings or on the bottom four strings. Learn how to transform the Maj7 voicing into other chord types: 7, min7, etc.
Taking the drop-2 CMaj7 in first inversion, the 3rd is on the fifth string, and the 7th is on the fourth string, so you can adjust to create drop-2 C7 and Cmin7 in first inversion:
An alternative method
Instead of starting from a close voicing in root position to get a drop voicing and then moving the notes of that voicing around to find the inversions, you can start from the inversions of a close voicing. That is, you can start by constructing a close-voiced CMaj7 as we did above, then finding its inversions by moving the bottom notes up an octave. Once you have each of the inversions, drop the necessary notes to get all of the drop voicings. For CMaj7, starting with a close-voiced C-E-G-B, you have the inversions: E-G-B-C, G-B-C-E, and B-C-E-G. These respectively become the drop-2 voicings: B-E-G-C, C-G-B-E, and E-B-C-G. These are the same voicings that we had before.
This alternative method is a great way to think about drop voicings, but not a great way to work with them on the guitar. The fundamental issue is that close voiced chords and their inversions are difficult to play on guitar. For me, it is easier to work out fingerings using the first method, but it is good to know about both.