OK, to make it nice and simple...
International cricket (for the team picked by the ECB-appointed selectors)
First-Class (county First-XI) cricket (and for some inexplicable reason UCCEs are allowed to play this First-Class stuff against the counties in a few games per season)
County Second-XI & Minor County (IMO this should be called "Second-Class" and did used to be) cricket
Club First-XI cricket (eg Exeter Cricket Club First XI)
Club Second-XI cricket (eg Exeter Cricket Club Second XI)
Club Third-XI cricket (eg Jeff Stanyer's XI - formerly known as Exeter Cricket Club Third XI)
Club Fou...
You get the picture?