The tooth whitening systems available at Eckerds, Rite-Aid, CVS and probably Walgreens, too, are essentially the same active ingredient: carbamide peroxide. Carbamide peroxide is a more stable form of hydrogen peroxide, which is what does the whitening (oxidation), and the solution is usually anhydrous glycerin based to give a thick gelled consistency.
When the gelled form mixes with your saliva, it breaks down (probably not technically 'correct' for all the chemistry brains) into hydrogen peroxide. The carbamide form is expressed in a percentage that is approx. 3 times the percentage of hydrogen peroxide it will yield. IOW, 20% carbamide will effectively yield 6.6% hydrogen peroxide.
Dentists have for decades recommended hydrogen peroxide as an oral rinse to promote good oral health. When hydrogen peroxide was available in strengths as high as 20%, people were using it straight out of the bottle without diluting it despite the dentists warnings. In this strength, it is extremely irritating to gums and can be caustic (concentrated peroxide is used in industrial cleaning processes, rocket fuels, and explosives). Today, it is only sold in 3% concentrations OTC.
The only advantage a dentist has is they are able to buy and use stronger concentrations of the same active ingredient. The dentist avoids irritation and potential damage to gums and oral tissues from these strong peroxide concentrations by forming custom mouth trays that "fit" your teeth to prevent the peroxide from contacting your gums except for some contact with the gum line. You want to bleach your teeth, not your soft tissues.
The consensus seems to be, higher concentrations + fewer treatments produce a better whitening effect than lower concentrations + more treatments. But, greater degrees of irritation and tooth sensitivity are more likely with higher concentrations, too, so there is a bit of trade-off here. A sensistive tooth toothpaste such as Sensodyne can be used to counter tooth sensitivity, though the senstivity will begin to subside on its own after a few days of terminating treatment.
I recommend products containing no more than 10 - 15 percent carbamide peroxide. Do NOT use this stuff if your teeth and gums are not in good condition. If you have a chipped tooth or a small cavity, this stuff can send you up the wall when that peroxide finds the nerve. It won't "harm" the nerve, but it will stimulate the heck out of it for 5 or 10 minutes. Don't say I didn't warn you.