A couple of points. Firstly, there's not a firm definition of a rescue - anyone can call themselves this, and there is no licensing for the sector. A pound is where a local authority holds stray dogs - it either is run by the Council itself or by a contractor. Getting a dog from a pound is a pretty cursory process - you turn up, pay your money and basically that's it. Not recommended unless you really know what you are doing, especially if you have DC, or are a first time dog owner. There is virtually no back up, no home check and little behavioural assessment.
A rescue can be "all-breed" or a single breed/type. Typically, a good all breed rescue (Hope Rescue is an excellent example) will take dogs from pounds or owner surrenders, keep them in foster homes for assessments and health checks and then make them available for adoption. This sort of rescue will usually have clear policies on their website about who they will adopt to (e.g. rules about minimum ages of children are typical, working hours of owners etc.) and how you can go about the adoption process. You will usually receive a home check which is to check your home is physically suitable for the dog you want, and that you understand the responsibilities of dog ownership. You then pay an adoption donation and the dog comes to live with you, but the rescue will often do a follow up phone call/visit to see how you are gettting on, and a good one will behavioural support and advice available as well as more general advice on caring for/settling in your new family member.
Single breed rescues will have a great deal of expertise in their particular breed and any issues you may need to consider. For instance, there are many greyhound rescues which find homes for retired ex racers. Similarly there are specialist rescues for lurchers (sadly there is a plentiful supply of both these types of dogs) and collies. Many of these dogs may not have come via a pound, but possibly direct from a racing kennel or working home, so a great deal is known about their breeding, training, background etc.
I'd be wary of any "rescue" that only has puppies - that's basically just a wholesaler trying to con people. Not insisting on home checks is a big red flag. They should be clear about where most of their dogs come from - are they poundies, or imported for example? They should offer good advice and support - run very fast from any rescue that advises you about being "alpha dog", or talks about "dominance". Most reputable rescues insist on dogs and bitches being chipped, vaccinated and neutered (unless contra indicated by a vet) - however this should be done at the appropriate age. They should be honest about any health issues or behavioural issues the dog has and should be able to say if the dog has been assessed in a foster home environment.
Bigger rescues are members of the ACDH, and most beyond the very smallest are registered charities. Personally I would steer clear of larger ones that are not charities - there is no good reason for them not to be one. Good ones take their governance seriously - look for transparency over accounts, meeting minutes, organisation structure etc. Also ask about campaigning. Anyone can rehome a dog - but what are they doing to prevent/minimise problems at a more strategic level?
Finally, follow up - good rescues will have a flourishing network of volunteers, and when you adopt from them, you will feel like you have joined a very large "family". Look for social activities, dog shows, walks, picnics etc especially if you have adopted a dog with particular quirks or specific breeds. Sighthound rescues are usually brilliant at this sort of thing.
Apologies for the essay!