The EU is the formal club of 28 countries, which will become 27 after Brexit.
The single market is the name given to those countries plus others such as Norway which trade freely with each other without tariffs, quotas or taxes on trade. It also requires free movement of goods, services, capital and - crucially - people. This means members of the single market must allow citizens of other countries to come and live and work.
The customs union is larger than the single market - it includes Turkey. It doesn't cover every area of the economy (eg not agriculture) but allows free trade within the union and agrees tariffs to countries outside it. From a Brexit point of view, it would allow the current soft border between Northern Ireland and the Republic of Ireland to continue. But it would mean the UK could not negotiate its own trade deals with other countries, eg China.
More info: BBC News