Above is a nice, simple and short chart of the various brainwave ranges and the types of effects they generally have and what sorts of activities they may be able to enhance. There simply is nothing more detailed than what this chart contains about entrainment frequencies and what they are useful for. Those other elaborate lists to be found on the internet are nothing but the product of overactive imaginations.
When two pure tones of slightly different frequencies are delivered simultaneously to the two ears, is generated a beat whose frequency corresponds to the frequency difference between them. That beat is known as acoustic beat. If these two tones are presented one to each ear, they still produce the sensation of the same beat, although no physical combination of the tones occurs outside the... [Show full abstract]
This is not to say that entrainment cannot produce some physical effects, particularly negative ones. It is possible to induce epileptic seizures using entrainment, especially visual entrainment that employs flashing lights. That’s why people with seizure disorders are uniformly told to consult with their health care provider before using entrainment if they have a seizure disorder or some other serious medical condition.
When stress overwhelms your nervous system, your body is flooded with chemicals that prepare you for “fight or flight.” This stress response can be lifesaving in emergency situations where you need to act quickly. But when it’s constantly activated by the stresses of everyday life, it can wear your body down and take a toll on your emotional and physical health.
Today, EEG machines are used for diagnosing epilepsy and sleep disorders, for determining dosages for anesthesia, and measuring the brain activity of people in comas or suffering from brain trauma. EEG machines also continue to play a role in researching and understanding brainwave entrainment and developing new and better methods for delivering the benefits of this form of brainwave modification. 
It might seem obvious that you’€™d know when you’€™re stressed, but many of us spend so much time in a frazzled state that we’ve forgotten what it feels like when our nervous systems are in balance: €”when we’€™re calm yet still alert and focused. If this is you, you can recognize when you’re stressed by listening to your body. When you’re tired, your eyes feel heavy and you might rest your head on your hand. When you’re happy, you laugh easily. And when you’re stressed, your body lets you know that, too. Get in the habit of paying attention to your body’€™s clues.
This music encourages a state of alpha relaxation. The alpha state is a pleasant state of relaxed alertness. It’s a state that many people experience when they are waking up in the morning or when they are just beginning to drift off to sleep at night. While in a state of alpha relaxation, the mind is quite clear and receptive to information, learning is accelerated and one’s sense of creativity is enhanced. The mind is also very open to positive suggestions during this state.
The Neuro Programmer website, as is typical, is full of testimonials and vague references to research. I tried to track down the research they referenced, but could not find anything published or peer-reviewed by the authors named or on the subjects indicated. For example, they cite one study by Thomas Budzynski, but the only thing published by him that is listed on Pub Med is a small study on biofeedback published in 1969. The 1999 study they reference is not listed. Also – Budzynski has connections to another company – Theta Technologies – which sells similar technology.  (The exact nature of his connection is not clear, he was at least a consultant. He is also listed as the director of research for another company, Synchromed, LLC.)  This looks like just another in-house worthless study to support the marketing of a product.
♥ At 56, I have tried many a thing. The default setting of Binaural Beat Machine does make me very relaxed. But it's not what it does on the moment, pleasant as this might be. It is the quality of the sleep, when I use it for half an hour before going to bed. I sleep a deep, restful sleep. I use it during day-time meditation as well, and the clear-mind feeling about an hour later is wonderful!
Binaural beats are the playing of two sine waves of close, but not the same, frequency into different earpieces of a set of headphones. The "beat" is heard as the brain tries to process them as a distinct sound, while taking into account the constructive/destructive interference. It is said to migrate brain waves to be closer to that frequency, like a sort of biological resonance, which is true. Of course the same can be said for listening to relaxing music help you relax, up tempo music to help you feel energetic, etc...
Building on guided imagery, you can also imagine yourself achieving goals like becoming healthier and more relaxed, doing well at tasks, and handling conflict in better ways. Also, visualizing yourself doing well on tasks you’re trying to master actually functions like physical practice, so you can improve your performance through ​​visualizations as well.
So, in summary, binaural beats certainly do not work the way the sellers claim, but there's no reason to think they're any less effective than any other music track you might listen to that effects you in a way you like. If they make you sleepy (like they all do for me), use them to go to sleep. If they relax you or get you amped, use them for that. But don't expect them to be any more effective than regular music. If someone you know claims that they are, put them to the test, and bust the myth.
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]