Throughout my learning process I encountered and used more than a few books, some of which I forgot their titles. Among those I still remember:

Arabic Course, by Dr V. Abdul-Raheem

This is the book they use at the Institute of Arabic Language, Islamic University, Medina, to teach Arabic grammar to non-speakers of the language. It comes in three volumes, and will give you a solid understanding of the Arabic language. After completing it you will have a really sound knowledge of the language.

As the book teaches Arabic through Arabic itself, you will definitely need a tutor. But if you study it after learning from other books I dont think you will have major problems. I highly recommend studying it from the beginning even after completing other books that teach Arabic using other than Arabic as medium.

Arabic made easy, by Abul Hashim

This is the book which is responsible for my great leap from beginners level to advanced level, in around two months. But then it was during school holiday. When I finished with it, I was able to read magazines and similar materials with considerable ease without the help of a teacher. I have yet to come across a book which will enable you to make so much progress in such a short time.

But the problem is that the book uses a very rigid style, purely grammatical, which will look dry to many. So be sure to have a good look at it before you go for it. However, if you feel comfortable with it, then it is a book you must use.

It is small and concise. You won't learn a lot of words from the book, but it will show you how to make the most of the small number of words you know.

A new Arabic Grammar, by Haywood & Nahmad

This is a nice book, with lots of explanation, vocabulary and exercises, covering almost all aspects of Arabic that a beginner has to learn. But if your mother tongue is not English, you might find the English used in the book difficult, which I do as I am no native speaker of English.

There is a mini-dictionary at the end of the book which will come in handy very often.

Recently i came across a book by the title "Arabic for Beginners", by Dr Syed Ali. It is also a nice book.