I have usually come across 終わる and know when to use it. But when I learnt about 自動詞　and 他動詞 I was surprised to see that　終わる　is actually intransitive. Thus I am confused now.
Why do we say
私は授業が終わりました. I have finished the lesson.
私は宿題が終わりました. I have finished the homework.

Rather than
私は授業を終えました. I have finished the lesson.
私は宿題を終えました. I have finished the homework.

Since WE are doing the action 'finish' on something else. Why not use 終える than? What is the correct translation of 私は授業が終わりました than? Since both sentences cannot have same meaning, i.e the one with 終わる and the one with 終える. I have only seen the word 終わる being used all the time.

You can certainly find examples of ~を終えました --

娘、七歳の七五三を無事終えました. My daughter safely got through her "shichi-go-san" celebration as a seven-year-old.

病院は営業を終えました. The general hospital has closed.

Japanese has a lot of transitive-intransitive pairs where in English we would definitely use the transitive, but in Japanese you can use the intransitive.

かたづく・かたづける　(to clean up) -- you can say 部屋がかたづいた 'the room straightened up' instead of 部屋がかたづけられた 'the room was straightened up' (by someone).

消える・消す
止まる・止める
届く・届ける　

There are a ton of these in Japanese, and a lot of them actually work in English more or less the same way, except that we only have one verb instead of two. If we say "The car stopped," normally we assume that someone put on the brakes, or if we say "The light went out" normally we assume that someone turned it off.

So, you can say 授業を終えました, or you can say 授業が終わりました. Keep in mind, the subject of 私は授業が終わりました is actually 授業, even though you're the one who finished the class. What you're actually saying, in grammatical terms, is 'As for me, class is finished.' (And you can omit 私 and say 授業が終わりました, which is probably the more natural way to say it).

うぅわあー。 どうもありがとうございました。

Yeah, I think you're just not used to the fact that English has just subjects, but Japanese has both topics and subjects.

私は授業が終わりました.
Here, は marks the topic, so the topic is 私.
が marks the subject.

終わる in English is 'to be finished' (just 'to finish' can be used without an object in English, but it's not as common as 'be finished').
So in English we say 授業が終わりました all the time--"The lesson is finished." Like English, there is just a subject here, so we can translate into English very easily.

However, in English we can't realy say 私は授業が終わりました so straightforwardly.
It means roughly, 'As for me, the lesson(s) is(/are) finished."
'Me' is the topic, but 'lesson' is the subject. Actually, the nuance is a bit different than 'I have finished the lesson', as it can have a wider range of meanings. For example, this could commonly be used as 'I don't have any more classes this semester.'

actually when you stick the "be" verb into the translation like that, you're making a passive construction. 終わる is not passive.
授業が終わりました doesn't mean the "lesson is finished". It means "the lesson ended".
終わる intransitive
終える transitive
終えられる passive (be finished (by someone))
終わっている be finished (in the intransitive sense, ie. "to have ended")

yay for nitpicking

That's not nitpicking; the difference between passive and intransitive is very important, especially in Japanese where the passive is used in ways that don't match with English.

It's nitpicking in the sense that I knew about it (in fact, I considered mentioning it), but I ignored it because ultimately I wanted to focus on how Japanese has both topics and subjects.

Nevertheless, it's good that you guys pointed it out to make things clearer in the long run =)

The big problem I think is that students are kind of taught that 終わる basically means end then assume that it means that in every familiar English context. Then when they start to learn 他動詞 vs. 自動詞 they discover 終える. In a lot of cases neither are really what students want to say. A lot of times 済む would be a better verb, but it has a "higher level" kanji so it's avoided so as not to "confuse" students.

I'm still not clear as to what the fine differences are myself, but I know that I hear 出来た！ and 済んだ at work a lot more than I hear 終えた.

Wasn't there a online quiz that you could practice for these questions. I saw it somewhere...

どうもうありがとうございました。