As is pretty clear, this has influence not just from the Tolkien scripts but from Arabic as well.

Non-final consonants almost always take vowels. Exceptions: N/L/R can precede other consonants. Words can end in any consonant or vowel.
Alphabet reads horizontally right to left.
In theory the language can be written without the vowel diacritics but they are usually marked. There is no marker for a lack of vowels. Short vowels are marked with diacritics. Long vowels are written as diacritics on a seat. The seat is also used for vowels following other vowels or initial vowels. When in doubt, the seat is a long vowel. Note, there are a couple of mistakes re: seats on the scanned page above. It functions like Alif in Arabic, which is to say initial and medial Alif doesn't connect to anything following it and neither does the vowel seat of Desert Elvish. It only connects to preceding letters. The only case in which and initial or medial seat connects to a following letter is if the following letter is another seat. Consonants are doubled with the symbol shown at the bottom of the diacritic column.
The language is written without spaces between words, only between sentences. There is little in the way of punctuation though quote marks and periods see some use.

Consonants:
Z/z (z)
V/v (β or v)
F/f (ɸ)
Zh/zh or J/j (ʑ or ʒ)
Kh/kh or X/x (x or χ)
Gh/gh or G̣/g̣ (ɣ or ʁ)
N/n (n)
R/r (r)
B/b (b)
L/l (l)
Dh/dh or Ḍ/ḍ (ð)
D/d (d)
Ṣ/ṣ or S/s (sˤ)
W/w (w)

Vowels:
Long:
É/é (ɛ:)
Â/â (æ/a:)
İ/i̇ (i:) 
O̗/o̗ (o:)
U̬/u̬ (ɯ:)
Short:
E/e (ε)
A/a (a)
I/ı (ɪ)
O/o (o)
U/u (ʊ)