How can a non-French can get a French citizenship?

There are two main ways of obtaining French citizenship for a foreigner.
The first one is by marrying a French national - you are then granted the right to apply for French citizenship (after some time, the French authorities will ask you to prove that you really live with your spouse)
The second common way is that France may accept applications from foreigners having spent legally a lengthy period of time (more than ten years) on French soil. After a criminal record check and the completion of a dossier, the applicant may have to demonstrate that he is well integrated into mainstream French society. French language abilities and the approval of French 'values' may be tested (recently an applicant was famously denied French citizenship because he didn't accept the idea of equality between men and women).

French government websites will explain these possibilities at length.