In both the UK and USA a county, in political terms is a second-level unit of sub-national government. In the UK the normal hierarchy goes something like
In both cases, power is delegated to counties from above, and the powers, services provided, etc. vary depending on location. The biggest difference is that the US tends to treat its "things not called counties" as functionally identical, whereas there's slightly more variance in how the non-counties fit into the UK governmental structure.
You mention county cricket in your question, which brings up one complication. Due to population and political change, the UK also possesses "historic" counties, and "ceremonial" counties. The former are Norman toponyms (i.e names for places) which used to be counties exerting governmental control, but may now no longer exist (one, Middlesex has a cricket team, despite its land having been taken up into modern London) while the latter represents the domains of the Lord Lieutenants, the avatars of the monarch in each region, who once held real political power, but are now just titles.