If your allergic to a cat or dog the proteins in their saliva do cause allergic reactions but more likely it's the dander that is bothering you. Dander in dogs is just like dandruff in people. It's flakes of dead skin and that's generally what people are allergic to.
It really depends on the air purifier you purchase. Some upright stand up types can run into the hundreds if not thousands of dollars where as you can also get bookshelf purifiers for as cheap as $20. Some work better than others.
I recommend looking a the Oreck brand of air purifiers. They have always had good quality products in my opinion and is worth the extra money in maintenance and in saving on filters. Buying filters can be expensive so consider the price of replacement filters when considering your purchase.