For example, if you do a home-stay, you should pull your weight around the house. This means that you should do work around the house (wash dishes, vacuum, so on) so that you are not adding too much work to family. If you are not pulling your weight it means that other people have to do more work because you are not doing work.
At a job, pulling your weight means taking care of all of your responsibilities. If you don't pull your weight at work, again, it means that your co-workers have more work because you are not doing enough.
So, you can use it like this:
A: Why are you working overtime?
B: I'm just trying to pull my weight at the office.
A: You're so lazy!! You need to pull your weight!!
B: I'm sorry, I'll work harder.
A: (to your host mom) What can I do to pull my weight?
B: How about the dishes?
A: I'm on it!! (I'll do it)
Hope that helps!! :D