A study by Tina Huang, PhD, and Christine Charyton, published in the September 2008 issue of the journal, Alternative Therapies examined the results of twenty previous studies measuring the effectiveness of brainwave entrainment for improvements in cognitive dysfunction and deficits, stress reduction, pain management, migraine and headache control, pre-menstrual syndrome, and behavioral difficulties, and all showed significant improvement in symptoms using entrainment techniques.
Rather than discontinuing my meditation practice altogether, I instead began researching methods of increasing the effectiveness of meditative practice. This seemed an inevitable decision for me in light of the undeniable improvements I had witnessed in response to previous experience with meditation. This resulted in my eventual re-discovery of brainwave meditation programs brainwave synchronization. I had used a few different such products previously, but none that seemed effectively designed to meet my specific needs. After deciding to try a particular brainwave meditation program, I found it to greatly assist me in producing all of the previously experienced benefits of meditation, but in a shorter period of time and in a much more focused manner.
So to summarize their claim, they're saying that entrainment means that a binaural beat will cause your brain's electroencephalogram to match the pattern of the phantom beat. Well, if it did, entrainment certainly doesn't apply and would not be part of the equation, so we can scratch that off the list. But it doesn't make the claimed observation wrong. We do know that certain electroencephalogram waveforms are often associated with certain kinds of activity. For example, physical activity or REM sleep often produces an electroencephalogram with a sine wave of between 4 and 8 Hz, which we term a theta pattern. Waking relaxation with eyes closed often produces a pattern from 8 to 12 Hz, which is called an alpha pattern. There are only a few characterized patterns, and pretty general descriptions of what kinds of activities go with them. The claim made by the binaural beat sellers depends on much more granular and specific matches. For example, the claim that a binaural beat with a frequency of X produces the same effect in your brain as Vicodin is wholly implausible. Such claims presume that we know the exact frequency of the electroencephalogram in each of these desired conditions, and the fact is that brain waves don't work that way. It is wholly and absolutely implausible to say that desired brain condition X will occur if we get your EEG to read exactly X Hz.

The Frequency following response (FFR), also referred to as Frequency Following Potential (FFP), is a specific response to hearing sound and music, by which neural oscillations adjust their frequency to match the rhythm of auditory stimuli. The use of sound with intent to influence cortical brainwave frequency is called auditory driving,[39][40] by which frequency of neural oscillation is 'driven' to entrain with that of the rhythm of a sound source.[41][42]
Resonant entrainment of oscillating systems is a well-understood principle within the physical sciences. If a tuning fork designed to produce a frequency of 440 Hz is struck (causing it to oscillate) and then brought into the vicinity of another 440 Hz tuning fork, the second tuning fork will begin to oscillate. The first tuning fork is said to have entrained the second or caused it to resonate. The physics of entrainment apply to bio-systems as well. Of interest here are the electromagnetic brain waves. The electrochemical activity of the brain results in the production of electromagnetic wave forms which can be objectively measured with sensitive equipment. Brain waves change frequencies based on neural activity within the brain. Because neural activity is electrochemical, brain function can be modified through the introduction of specific chemicals (drugs), by altering the brain's electromagnetic environment through induction, or through resonant entrainment techniques.
The Transparent Corp forum – This forum is an invaluable resource for any brainwave entrainment user or enthusiast.  Most of the feedback is obviously focused on the Transparent Corp software, but with over 20,000+ posts now you can find answers to the whole array of brainwave entrainment questions on there.  (UPDATE: Sadly, the Transparent Corp forum has now been taken offline)
When signals of two different frequencies are presented, one to each ear, the brain detects phase differences between these signals. "Under natural circumstances a detected phase difference would provide directional information. The brain processes this anomalous information differently when these phase differences are heard with stereo headphones or speakers. A perceptual integration of the two signals takes place, producing the sensation of a third "beat" frequency. The difference between the signals waxes and wanes as the two different input frequencies mesh in and out of phase. As a result of these constantly increasing and decreasing differences, an amplitude-modulated standing wave -the binaural beat- is heard. The binaural beat is perceived as a fluctuating rhythm at the frequency of the difference between the two auditory inputs. Evidence suggests that the binaural beats are generated in the brainstem's superior olivary nucleus, the first site of contra-lateral integration in the auditory system (Oster, 1973). Studies also suggest that the frequency-following response originates from the inferior colliculus (Smith, Marsh, & Brown, 1975)" (Owens & Atwater, 1995). This activity is conducted to the cortex where it can be recorded by scalp electrodes.  
Over the centuries people have discovered that repetitious rhythms such as the beating of a drum at a single tone, or chanting, or other methods for producing a single, repetitious rhythm, have been known to produce certain states of consciousness whether deep relaxation, trance, a state of meditation and so on. The rhythms used basically changed the brainwave state of the person listening as long as that person remains open and receptive to the rhythms.
The composer Alvin Lucier has written many pieces that feature interference beats as their main focus. Italian composer Giacinto Scelsi, whose style is grounded on microtonal oscillations of unisons, extensively explored the textural effects of interference beats, particularly in his late works such as the violin solos Xnoybis (1964) and L'âme ailée / L'âme ouverte (1973), which feature them prominently (note that Scelsi treated and notated each string of the instrument as a separate part, so that his violin solos are effectively quartets of one-strings, where different strings of the violin may be simultaneously playing the same note with microtonal shifts, so that the interference patterns are generated). Composer Phill Niblock's music is entirely based on beating caused by microtonal differences.

Binaural beats were first discovered in 1839. The effects of using audio instead of light for brainwave entrainment were noted in 1959. Oster’s research paper in 1973 brought attention to the use of binaural beats in a medical setting and this inspired a range of studies in the 1980s indicating very positive findings. The effects of brainwave entrainment using binaural beats has become of a hot topic of research ever since but despite their reported successes, they have never quite shook off the sometimes negative connotations of being an “alternative medicine”.
I am fortunate to be working with Deepak Chopra, M.D., and Dr. Rudy Tanzi, co-authors of the bestselling book Super Brain, on a technology called Brain Wave Entrainment. Deepak is very well known, but Rudy is an amazingly interesting person as well. He is the Kennedy Professor of Neurology at Harvard Medical School and Vice-Chair of Neurology at Mass General Hospital. Rudy co-discovered three of the four original Alzheimer's genes and today runs the Alzheimer's Genome Project. He also plays the keyboards, including, at times, for Aerosmith. He is kind of a real life "Buckaroo Banzai."
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A more extensive study of over 100 participants who were undergoing general anesthesia for a day procedure, reported a decrease in pre-operative anxiety. The participants in this study listened to 30 minutes of binaural beats before surgery, but the researchers noted that people experiencing high levels of pre-operative anxiety could listen to binaural beats for up to 1 hour before anesthesia to reduce levels of anxiety.