OPTION A: Today, I finished myone task for jobat work.¶
OPTION A: Today,I finished one work task .

but,other tasks don't finish because my boss can't finished giving tasks for meHowever, I haven't finished other tasks because my boss keep piling new tasks on my desk.

He didn't come our workplaceToday, my boss didn't appear at work.

OPTION A: I don't know the reason.¶
OPTION B: The reason for that is unclear.

Today (11/24)

Today, I finished my task for the job.

I am very happy!

bBut, other tasks don't finish because my boss can't finishedkeeps giving tasks forto me.

I don't know the reason.

Today, I finished mya task for my job.

I'm very happy!

In English, we can't omit the subject the way Japanese does.

bBut, my other tasks doaren't finished because my boss cahasn't finished giving those tasks forto me.

He didn't come to our workplace.

I don't know the reason. (or: I don't know why)

Today, I finished my tasks for my job, which made me very happy.

I am guessing that there were multiple tasks. Also, the job is yours, and hence you should have "my" before it.

very happy!

"Very happy" is not a full sentence. Instead, you should add it to the previous sentence.

but,However, other tasks don't get finished because my boss can't finished giving tasks for mekeeps giving me work.

Start each sentence with a capital letter.
Also, instead of using "But" as the start of a sentence, it is better to use "However".
Other than that, most of the sentence was correct grammatically, but didn't flow well.

He didoesn't come to our workplace.

I don't know the reason why.