There is no single answer to the question of how much protein to consume daily. The answer depends on activity status, health status, pre-existing conditions (either medical problems like renal disease, or lifestyle choices like vegetarianism) and, even with all the above considered, a range is more viable than a set number. Use the following as a guide.
If you are an athlete or highly active person currently attempting to lose body fat while preserving lean muscle mass, a daily intake of 1.5-2.2g/kg bodyweight (0.68-1g/lb bodyweight) would be a good target.
If you are an athlete or highly active person, or you are attempting to lose body fat while preserving lean mass, then a daily intake of 1.0-1.5g/kg bodyweight (0.45-0.68g/lb bodyweight) would be a good target.
If you are sedentary and not looking to change body composition much, a daily target of 0.8g/kg bodyweight (0.36g/lb bodyweight) and upwards would be a good target.
Two other things should be noted as follow-up:
Studies have only really intensively looked at dosages up to 1.5g/kg bodyweight, and others have touched down on dietary intakes in the 2.2g/kg or 3.0g/kg range. That being said, there do not appear to be any apparent negative effects to a higher protein intake. See this FAQ topic for more information.
If you are obese, using a protein intake relative to body weight is a bad idea. Either calculate your lean mass (overall weight after subtracting fat mass, which can be calculated by body fat percentage) or use your goal/target weight for calculations.
This is usually the lowest recommended estimate as it does not assume any extraneous conditions. It may not be sufficient for elderly persons undergoing the process of muscle loss, as inadequate amino acid intake can result in muscle mass loss to mobilize those amino acids for other uses at this level of intake.
Two studies indicated that 12-15% of calories from protein is sufficient for active individuals (60-75g of protein for an individual eating a 2000 calorie diet). While two more recent studies  argue for higher intake - the first study reports that greater than 1.6-1.8 g/kg of bodyweight (0.7-0.8 g/lb of bodyweight) may be necessary, while the second study indicates that up to 3.0 g/kg of bodyweight (1.4 g/lb of bodyweight) isn't harmful, and may have additional minor benefits.
According to the International Society of Sports Nutrition, protein intakes of 1.4-2.0 g/kg of bodyweight (0.6-0.9g/lb of bodyweight) for physically active individuals is not only safe, but may improve the training adaptations to exercise training. . The American Dietetic Association, Dietitians of Canada, and the American College of Sports Medicine also support high protein intake for active individuals in the range of 1.2-1.7 g/kg of bodyweight (0.5-0.8 g/lb of bodyweight).
The reasons for the above tend to be increased leucine oxidation (a marker for amino acids being used for fuel, by being turned into glucose) that requires a higher intake of amino acids to negate and preserve nitrogen balance. Additionally, increasing protein intake above the previously defined RDA 'daily allowance' will increase protein synthesis and, at levels higher than double this total, decrease protein breakdown. Increased muscular hypertrophy is seen as beneficial to sports performance.
During Weight Loss
High protein diets have been found to preserve lean body mass when dieting in both obese people and athletes  and has also been shown to improve overall body composition. A doubling of protein intake from 0.9g/kg (near the daily recommended intake for the general population) to 1.8g/kg is able to preserve lean muscle mass during short-term and relatively drastic drops in calories.
In one of the stranger studies, males eating 3.0 g/kg of bodyweight (1.4 g/lb of bodyweight) improved reaction times more so than those eating 1.5 g/kg of bodyweight (0.7 g/lb of bodyweight), over a 3 week period.  This may be due to higher levels of neuroactive amines such as L-Tyrosine, DL-Phenylalanine and L-tryptophan; Phenylalanine was recorded in higher levels in the blood, and the branched chain amino acids recorded in the blood beneficially influence L-tryptophan and serotonin pharmacokinetics.
Other benefits of protein can be found on our whey protein page.