Dander is material shed by animals, mostly skin cells but also hair. Many people with allergies are also allergic to animals, reactions can involve the eyes (conjunctivitis), nose (rhinitis), lungs (wheezing) or skin (rashes or eczema).
Clapping/stroking the animal will cause the worse reactions, but you may find you react to touching surfaces (especially carpets, soft furnishings) where animals have been, or even to dander in the air.
Cats - if you’re allergic to one cat, you’re likely to be allergic to all cats! The same is not necessarily true of dogs. Cat dander is particularly sticky, it can be found in houses years after a cat has lived there, and it clings to clothes so you may react to being in close contact with somebody who has been in a house with cats.
Dogs - the idea that some dogs are “hypoallergenic” or “low allergy” is popular, but not based in much science! It is true that you can be worse with some dogs than with others, but this is often as much to do with how often and well groomed the dog is, as it is to do with the breed. You may also become tolerant (ie not react as much) to an animal that you are in regular contact with. In fact, there is good evidence that having a dog at home protects children from allergies in later life!
Testing - because everyone is different, allergy testing for pets is of limited usefulness, and will rarely be reliable enough to justify removing a pet from the home, or to predict that a new pet will not cause any problems
Avoidance - don’t buy a new pet! Keep pets away from living and sleeping areas, and certainly do not allow them on sofas, beds etc. Brush regularly outside (every day), wash regularly (eg every week). Wash bedding and any other fabrics that have been in contact. With cats, wash walls too. Don’t let allergic children play on carpets where pets have been. People with cats and horses should ideally change their clothes before coming into the house of an allergic child.