B. It's the contraction of -에는. 

-에 is the particle that you should mainly use for time expressions.

토요일에 on Saturday

다음 주에 next week

십이월에 in December

내년에 next year

You can't just use -에는 for no reason, the added -는 part is the topic particle. In this case it adds the meaning of "as for", so there should be multiple things that you are "comparing".

-토요일에는 도서관에서 공부하지만 일요일에는 집에서 공부해요.

On Saturday I study in the library, but on Sunday I study at home.

토요일+엔 도서관에서 공부해.

1-토요일+엔 도서관에서 공부해.
2-그리고 일요일+엔 집에서 청소해.
3-오전에+는 도서관에서 공부해.
4-그리고 오후에+는 집에서 청소해.


A: What is 엔? Does it mean "on"? like on Saturday?

엔=에+는

토요일엔 = on Saturday(s)

B: Or is a combination of 는 and 에?

Yes, 엔=에+는

C: If both are true, then can I use it for all time expressions? I mean, for days, months and seasons so on.

Yes, you can use both expressions but normally we say 토요일에, 일요일에, 다음 달에, 오후에, 오전에..

When we say '엔' like 토요일엔, 일요일엔, it has some 'refusing nuance'.

Ex) A guy asks you out for a date, you want to refuse it politely. 

He asks: 토요일에 뭐해?

You answer: 토요일엔 도서관에서 공부해야 해.(means I'm not available, sorry)

He asks: 일요일은?

You answer: 일요일엔 집에서 청소해야 해.(means I'm not available, sorry)

Help me please.