In simple words Decidability of Membership Problem means that : "For a language L , given a string W we can always check whether W belongs to L or W does not belongs to L in finite amount of time i.e an algorithm exists such that given a string W it will always say YES or NO ".
Regular Languages : "Membership problem is decidable" .A Finite Automata can be drawn and W can be traced starting from initial state, if it reaches final state then W is member else not.
Context Free Language: "Membership problem is decidable": CYK algorithm exists
Reursive language: A language L is Recursive iff their exists a Halting TM M such that it accepts all W belong to L and rejects all W that does not belongs to L.thus "Membership problem is decidable"
Recursive Enumerable Language : "Membership problem is Undecidable". their exists a TM M such that it accepts all W belong to L and Halts but if W does not belong to L ,the TM M might not be able to reject and does not halt.
