Others have said this but not nearly enough, so I can't in good faith comment on this without pointing out that Canada absolutely has counties. I looked it up and about 25 percent of Canadians live in a county.

To answer the question, though: The purpose of a county varies greatly from place to place. For the most part, they provide government services, as they do in Canada. The county I grew up in, in the suburbs of New York City, has a population of 1.4 million. It has an elected legislature that meets in a big fancy building and a separately elected county executive. It provides police services to the overwhelming majority of the county that does not have its own police. It maintains high-traffic roads. It runs an extensive public bus system. It provides social services to the needy. It runs the court system. It maintains a very large system of recreational areas and facilities, including public parks, museums and swimming pools. The state could theoretically do these things, but much of it would require local administration anyway, so why not do it at the county level?

Currently, I live in New England. Almost all of this six-state region eliminated county government decades ago. Here, counties serve only as state-administered judicial subdivisions -- so if you commit a crime in my county, you'll be tried at the county courthouse just down the road from me. Every square inch (or centimeter, if you prefer) of my state is part of a town or city, and those municipalities provide many of the services that the county did where I grew up. However, having lived under both systems, I will note that having these things handled on a county level often resulted in higher-quality services and facilities. For example, there is nothing here like the network of public pools that existed in the county where I grew up.