Non-Terminating decimals - These are thedecimal number that goes on forever. 
While expressing a fraction in decimal form, when we perform division we get soon remainder. If the division process does not end, i.e, then such decimal numbers are know at Non-terminating decimals.
E.g : 2.666666

Terminating decimals - A decimal number that has a digit which do not go on forever is called terminating decimals.
E.g : 0.25

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Terminating decimals are those which which on expanding have a finite and countable expansion. That is those which end on a certain point or those which terminate. 
EX-  2.14 ; 5.8 ; 6.125

Non- terminating decimals are those that don't terminate but keep continuing. However non-terminating decimals have two types. 
i) those who's decimal expansion repeat themselves after a certain point, ie. non- terminating repeating decimals.
EX- 1.484848 ; 2.222222222... ; 3.146464646...

ii) those which are  non- terminating,non- repeating. ie. they don't repeat the numbers but don't terminate either.
Ex- 5.9156242387... ; 1.4570845... ; 23.56487545...