International cricket
County Cricket (1st & 2nd eleven) - the likes of Surrey, Lancashire, Warwickshire, Yorkshire, Durham etc.
Minor County Cricket - the likes of Oxfordshire, Bedfordshire, Devon
Club/League cricket - which is being discussed here
Club/league cricket is usually played by players who play cricket purely as a hobby on the weekend. The clubs usually sign 1 or 2 international players every summer, and usually they try and get players who are performing decently in domestic cricket in Australia, South Africa, Sir Lanka, Pakistan and occasionally India. Those international players will play for the 1st XI team of the club and also help train the younger players and will occasionally run training sessions.
Occasionally the clubs manage to scoop a big(ish) name - for instance the club I used to play for managed to scoop up Imran Nazir (if I remember correctly) in the early 2000's for a season (before I joined the club), or a club down the road from where I live managed to sign Sreeshant for this season, however his NOC got rejected by the BCCI and he couldn't make it.
A few years back - when I was at school - the club I used to play for managed to sign up a young promising up and coming Australian batsmen who was doing well in domestic cricket, however, Cricket Australia then didn't let him come to the UK to play league cricket because they wanted him to spend the off-season months training in Australia, just in case he gets a call up to the national side. His name was David Warner
So occasionally you do end up getting a few decent standard players in the club level teams, however, 90-95% of the players just play the game for fun. For players coming from the sub continent it is a decent summers income to support their families, as well as experience of paying in different conditions to what they are use to, which can never hurt.