The main reason why France clung to Algeria as long as possible was mostly people : french colonists that had settled in Algeria over time, what we called the "Pieds Noirs" (litteraly "Black Feet") Just before the independence, they were more than 1 million french citizens living in Algeria, which was administered as French territory (subdivided in departments, like metropolitan France, for instance, it was not a protectorate like Morocco or Tunisia..)

These "Pieds Noirs" had mostly French roots, but many of them were descendants of Italians or Spanish immigrants in Africa that had arrived during French colonization. Among Pieds Noirs were also many Sephardi Jews that had received the French citizenship through the decret Crémieux (1870), as explained in detail the post linked couple hours ago. The local Muslim inhabitants, on the other hand, the "Arabs", were merely "subjects", "indigènes" throughout the history of French Algeria. They became "French Muslims" after 1947, but never enjoyed equal rights or genuine self government until independence. (The "Arabs" will become the Algerians and also rather often consider themselves as much as Berbers as Arabs now, but I will not delve further into the question of Algerian identity)

This "Pied Noir" issue is so important that when president De Gaulle signed the Evian accords in 1962 granting independence to Algeria and ending the war, a quasi-civil war erupted between De Gaulle and a terrorist organisation known as the OAS (Organisation Armée Secrete, Secret Army Organisation, created in 1961) which represented the Pieds Noirs that were staunchly opposed to Algerian Independence, and felt betrayed. The OAS plotted to kill De Gaulle and almost succeeded on August, 22 1962. The OAS attacks took place on Algerian and French soil. A year before the Evian Accords, a military putsch against De Gaulle also took place, in April 1961 ; at least four generals seized power in Alger, so that Algeria remains French. The coup failed.

Interestingly enough, De Gaulle had also been brought back in power in 1958, because of the threat of another similar military coup, ending the 4th Republic and starting the 5th Republic, which is still France current constitution. (About this "coup" that greatly helped De Gaulle to make his come back in a slightly suspicious way, you can read Michel Winock, L'agonie de la IVe République : 13 mai 1958 I have no English reference at mind about that topic that would deserve volumes..)

Yet, in 1962, all these efforts were proven worthless. The violent actions of OAS, along with extreme brutality against "colonists" from ALN (The Algerian National Liberation Army) only hasten the departure of one million of Pieds Noirs civilians from Algeria to Metropolitan France, giving Algeria it's current ethnic makeup, which is a predominantly Muslim, Arab-berber country with almost no Sephardi Jewish (some left for Israel, most went to France) or "European" inhabitants. At that time "La valise ou le cercueil" , "suitcase or coffin" was a common saying among Pieds Noirs still living on african soil. Along with Pieds-Noirs, Muslims auxiliary troops that fought alongside French, the "Harkis" sometimes managed to escape to France, those who did not were often branded as traitors and massacred by the ALN.

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Economically, France also loses important assets : a well developed colonial agriculture, with substantial wine production to cater to the needs of the mainland, the best lands having been taken away from the locals. Also, France loses access to oil reserves, that are currently very important for Algerian economy (but at the time of independence were not such a huge asset, in a context of low oil prices). Finally, the desertic south of Algeria was also at the time being used as a testing field for various space rockets and nuclear weapons... However, those nuclear tests went on for five years after Algerian independence, by agreement between the French and Algerians. They are certainly not the main reason why France fought a bloody war against Algeria..

Please excuse my English (I'm French) and the current lack of sources. There is now quite ample French bibliography on the topic. I also omitted all the gruesome details of the conflict itself. It is not an attempt to hide the atrocities committed by both sides but I considered that aspect as slightly out of scope and lacked time.

If someone can provide English books on the topic, or more specifically extra input about US involvement on the matter (I think i read somewhere that they somehow helped Algeria to become independent) I would also be interested, though. There is probably a broader geopolitical "cold war" dimension to delve into.

I think there is still room for more information, but you will find two informative responses (from /u/thesoulphysician and /u/DerProfessor) to a very similar question on this post.