Both of these sentences can be acceptable:
- I finished my homework this week.
- I've finished my homework this week.
This seems rather confusing, but the subtle differences of the present perfect and the simple past can be seen by where each one can't be used: As both HostileFork and Ustanak stated, the present perfect is used when referring to a time period which includes the present whereas the simple past is used when referring to a time period which is "over". Therefore, notice:
- I went to Bangalore in 2009.
- *I've gone to Bangalore in 2009.
The second example above is unacceptable because, even though 2009 refers to a long stretch of time, it is still a single reference point in time which is now "over". Notice the differences in these examples:
- I went camping this week.
- I've gone camping this week.
Here, both are again acceptable, but the meaning is very different: In the first example, the speaker went camping and is now done with camping (They're "back" from camping); In the second, however, the speaker is actually still camping, e.g. they're telling someone on the phone that they're currently camping (and so e.g. they aren't available to go to the movies). Notice, again, that e.g. *I've gone camping last week is wrong for the same reasons that the example with 2009 is.
Likewise, notice where the simple past can't be used:
- I've played the saxophone for a year now.
- *I played the saxophone for a year now.
The second example here is unacceptable because the word now means that the time being referred to is ongoing and so, as explained above, the simple past should be used instead, as in the first example.