A Convenience Sample is a type of non-probability sample where the units have been selected because they are convenient for the researcher (which may not be the optimum sample for the research project). Typically Convenience Samples are considered 'bad' by the academic and research community writ large, but there are times when it is the only type of sample available. For example, if one is targeting a very low-incidence audience for survey (e.g. left-handed iPhone users with two or more pet ferrets), it is unlikely that the use of a probability-based or general public screening will be timely or cost-effective, or even successful at all. However, if we are able to purchase a membership list for the Left Handed Ferret Owners Society of America (made up of course), this allows us to have a much better chance of reaching people who meet our key criteria. However, it is a convenience sample, and undoubtedly the people who have joined this Society are different from those who meet the criteria but have not joined this Society, and one must take that into account when drawing conclusions from a survey administered to this audience. BUT, perhaps a 'convenient' list is preferable to no list at all. This is a judgement call by the researcher and client.
Online research involving panels is almost always categorized as 'convenience' sample (as opposed to randomized in some way, where everyone in the target population has an equal probability of selection for the survey). This is a fair categorization, but also doesn't inherently make it bad data... as long as the researcher thinks about and is fully aware of the implications of the sourcing for their sample, and factors this into their analysis and reporting, this kind of convenience sample can still be highly useful.
Situations where we most often encounter problematic convenience sample are when client's supply us with lists of their own. For example, a software company may wish to conduct a survey on how users of their software feel about the software. They will often send us a list of users they have, believing this to be a comprehensive list of users, or close to it. However, we must - as researchers - question the origin of these lists to assess how comprehensive they are: are they lists of people who signed up to receive software updates? This is likely to be a good list, because users of the software have an incentive to get the updates. We cannot assume it is perfect, but if users have an incentive to be on this list, it helps. Alternately, is the client's list from their newsletters, or from a list of people who have registered complaints about the software? In these cases, the list is far less likely to cover the full range of users of the software, and is clearly a more compromised 'convenience sample'. Beyond this, can the company's software be sold in other ways NOT via the company itself? If so, it is likely that the company cannot provide a comprehensive list of users at all, and we must begin to look outside our client for users (via other lists; screeners; etc).