I am a former Army Arabic linguist, and a graduate of DLI/FLC Monterey. This dictionary is the primary one we used, both at DLI and later on the job. An old Army buddy, I have a deep abiding affection for this thing, it being a true linguistic masterpiece and longtime companion.
Hans Wehr was professor at University of Munster from the fifties thru the seventies. This dictionary was first published in 1952 in German as "Arabisches Worterbuch fur die Schriftsprache der Gegenwart" - a mouthful of a title if there ever was one. We just called it the Hans Wehr. To my knowledge, it is the only dictionary that's organized to properly exploit Arabic morphology, which is to say the consonal root system.
Any other approach makes a hash out of the Arab language. Straightforward alphabetical ordering is ill suited to Arabic. The only traditionally alphabetically organized Arabic dictionary that I've seen (and I've seen quite a few) which is any good at all is the Lebanese Al-Mawrid. But I use it only as an occasional adjunct to Hans. Once you get a hang of Hans, and your vocabulary and sense of Arabic grows, the Mawrid will only be very occasionally useful from Arabic to English. It may not seem possible to a beginner (it certainly didn't to me,) but the Hans Wehr will come to make much more sense, and become much more accessible, than any other Arabic to English dictionary. So if you are new to this game, suck it up and use Hans as much as possible.
One sole caveat: for Modern Standard Arabic (MSA) the Hans Wehr is incomprable & indespensible. MSA is the modern universally written form of Arabic, and the pan-national lingua franca spoken in formal settings- on TV, in courts, etc., by educated Arabs. Note, though, that the local dialectical and classical forms of Arabic are very different balls of wax, and with them this dictionary can be of only limited application. I found it pretty wanting when dealing with Egyptian dialect while in Cairo, for example. For the Gulf, Jordan, Palestine and rest of the Levant it's a little more useful.
A brief grammatical explanation as to why the Hans is the only game in town (sorry if this is redundant or too didactic, but everyone should know what the deal is:)
As others have said here, the majority of the words in Arabic are derived from a three (usually) consonant verbal root. You take the consonants - e.g., KTB which denotes writing, or 'KH'RJ denotes outward movement - and vowel them with fatahs, or short 'a' sounds (which like all short vowel sounds are unrepresented in everyday writing) and you get KaTaBa, which basically means "he wrote" or KHaRaJa which means "he went out" - both the 3rd person masculine past perfect. This is used as the essential base form of the verb, equivalent to the infinitive in an Indo-European language.
By changing the unwritten "short vowels" (re-vowelling) and by adding other written "long vowel" and consonal sounds, you can generate a slew of words related to the verb in question. For example, by adding an M (mim) to the KTB (MaKTaB) you get "office." An A (alif)(AKTuB) gives you "I write." MaKTaBA = library. KiTAB = book. KHaRUJ = exit. KHRAJ = tax. KHiREEJ= graduate MaKHRaJ = place of exit. You get the idea.
Yet it gets crazier, and more beautiful, still.
By voweling this consonal root in different ways, and by adding other consonants as prefixes or suffixes, you place the verb in different measures. Now this is critical: measures are different forms the verb takes to change its meaning. They are one of the main reasons Arabic is such a zany, beautiful (some would say opaquely impenetrable) labyrinth of meaning. There are (I believe) 26 different measures in Classical Arabic. Just thank Allah that the modern Arabs have cut that number down to 10 in Modern Standard. Ten is wild enough. They are explained in the introduction of the Hans Wehr pp. xii - xv. Read these pages. They help explain the signifigance of all the roman numeralled entries in the dictionary.
All of the above means, of course, that an straight, western style alphabetical dictionary is nearly incoherent in Arabic. The fact that verbs all have all the same classes of prefixes, and the same patterns of internal suffixes & suffixes, not to mention complex patterns of conjugation in all the measures & moods, means that you have potentially hundreds of words for each of dozens of prefixes and patterns. And the voweling issue clouds things even further. That's not even to mention further insanity such as irregular nominal plurals, cases (nunation!) & other nominal suffixes (NB: beware verbal nouns.) Our Germanic/Latinate linear methodologies just don't apply very neatly. Better to approach the system as it's organized: radically.
Better to go from the heart from which all the meaning radiates, the root system, and classify words thereby. The roots are all listed alphabetically, but their relatives are all listed under them. Thus nouns like MaKTaB, MaKTaBA, KiTAB, and even more complex permutations such as AKTiTAB (registration) or ASTiKTAB (dictation), etc. will all be found under KTB on pp. 951 & 952 of Hans. Irregular plurals are of course listed under their singular form.
The nouns are just the appetizer, though.
The real piece de resistance are the aforementioned roman numeralled enteries of the main root entery. These give you the meanings of the measures. Not every verb is put into every measure- some are only in practice used in one or two. KTB is used in the I, II, III, IV, VI, VII, VIII, & Xth measures. Referring to the introduction pp. xii-xv, you see that PP 3rd masc. sing. measure I is KaTaBa, to write. KAaTBa is measure III, which means to correspond. VI, TaKATBa also means to correspond (get used to redundancy in Arabic, as well as *ahem* apparent contradiction.. You will learn to savour repetition & ambiguity!) Measure X, ASTaKTaBa means to dictate..
And so forth.
Abberations, cognates and foreign words - which ironically will probably throw you for a loop quicker than native Arab ones will - are listed in straight alphabetical fashion among the roots.
You have to begin to learn the morphology, learn to i.d. roots in your sleep, and internalize the patterns. Then pick up the Hans Wehr and let the fun begin!
At DLI one of our profs has created a "Conjugations of the Ten Measures" verb chart which is requisite to fully & easily utilizing this dictionary, especially for beginning and intermediate students. It apparently exists in no other place. I don't know why there isn't one in the Hans, seeing as how all those nutty German positivists have seemingly considered & fully classified nearly everything else. Try googling it, maybe you can dig it up online. Or else try to run a DLI Arabic alum down, and get a copy off him or her. If all else fails, you can perhaps construct a similar chart using the Hans (again, pp. xii - xv) and a good grammar book. I have my chart sealed to the back of my dictionary for easy reference.
This dictionary strips the patterns of the Arab language bare. This incredible semantic superstructure is one even many native speaking Arabs are mostly unaware of. Still, sometimes - okay, maybe often - this dictionary (like scientific things can so often do) kills the puppy, dissects it, and gives you a chart. But the essence of that puppy isn't in the chart. It's a more metaphysical reality. A living, breathing, wriggling thing, one that requires much more intuition than logic to understand.. The Arab language is just such a reality. Another semantic universe, truly exotic to an English speaking mind, a vivid poetic wonder.
You know, studying Arabic is a real pleasure. It's truly sublime.
Anyway, this is the dictionary you must own if you are in any way serious about Arabic. Using it can be a both exhilarating (no joke) yet exasperating and loopy experience, given how it represents the culmination of centuries of Orientalist scholarship.. Which means it is both a huge triumph, yet somehow simultaneously a subtle subterranean disaster. The Teutonic mania for clinical order and classification creasing into the envelopping cyclically anarchic mystery of the Semitic mind... Or something like that.
In any case, it is a masterpiece, which is good seeing as how it's apparently nearly all we've got.