A strapless heart rate monitor watch, as its name implies, does not come with a chest strap. For some people this is a great feature as you do not have to put up with the potential discomfort of wearing a chest strap (though once you get used to wearing it, it stops being particularly noticeable). Still, if you need to monitor your heart rate throughout the day, then wearing a chest strap all day isn't something that everyone looks forward to.
A typical strapless heart rate monitor has sensors on the face of the wrist watch. When you want to measure your heart rate, simply put two of your fingers on the sensors. After a few seconds, your heart rate appears on the display. This means two things: first, you can't measure heart rate continuously and second, you can't use it in high intensity exercises (when stopping to measure your heart rate for a few seconds is impossible).
Some strapless heart rate monitor watches are ECG accurate (i.e. they measure electrical activity of the heart and are thus very accurate) but many are not. Mio heart rate monitors are ECG accurate.
Those brandless ones that you can get on eBay for a few dollars are most likely to be non-ECG accurate. They use optical sensing technology to detect blood pulses going through your finger and so are not terribly accurate. Check carefully before you buy.
There is another type of strapless heart rate monitors called finger heart rate monitors. You wear a glove-like device with the pulse sensor wrapped round your finger to provide continuous heart rate monitoring. The two manufacturers that make this kind of HRMs are PowerGlove and Mark of Fitness.
Below is a compiled list of strapless heart rate monitors that have been featured on this web site. It is by no means an exhaustive list.
This shortened list is merely intended to provide the starting point of your investigation, especially if you are short of time. You should not rule out other monitors not on this list as they may well have other features that you will find very useful.