Houseplants do not have an appreciable effect on the concentration of volatile organic compounds (VOCs) in normal home environments: it would require between 10 and 1000 houseplants per square meter to achieve the same level of VOC removal as occurs from passive exchange between indoor and outdoor air. 

The idea that houseplants have an appreciable effect on indoor air quality is largely based on the Clean Air Study conducted by NASA in the 1980s. The study tested plants in sealed chambers and with highly elevated levels of volatile organic compounds (specifically benzene, trichloroethylene and formaldehyde). It concluded that to the extent that houseplants improve air quality, the mechanism of action is microorganisms within the potting soil, rather than the plants themselves.

Houseplants do have a statistically significant effect on the concentrations of both carbon monoxide and carbon dioxide in normal indoor environments, reducing carbon dioxide levels by 10-25% and carbon monoxide levels by up to 90%.  The effect has been investigated by NASA for use in spacecraft.

Plants also appear to increase the number and diversity of bacteria in homes, while reducing fungal diversity. They tend to increase humidity, reduce temperature swings, and reduce noise.

There are also many claimed psychological and physiological benefits to having houseplants. A critical review of the experimental literature concluded "The reviewed studies suggest that indoor plants can provide psychological benefits such as stress-reduction and increased pain tolerance. However, they also showed substantial heterogeneity in methods and results. We therefore have strong reservations about general claims that indoor plants cause beneficial psychological changes. It appears that benefits are contingent on features of the context in which the indoor plants are encountered and on characteristics of the people encountering them."
What the the effects of houseplants?
There are also many claimed psychological and physiological benefits to having houseplants. Houseplants do have a statistically significant effect on the concentrations of both carbon monoxide and carbon dioxide in normal indoor environments. 
Plants also appear to increase the number and diversity of bacteria in homes, while reducing fungal diversity.They tend to increase humidity, reduce temperature swings, and reduce noise. There are also many claimed psychological and physiological benefits to having houseplants. Indoor plants can provide psychological benefits such as stress-reduction and increased pain tolerance.