Weighty things are either very heavy or extremely important and worrisome. Putting your weighty cat on a diet might be a weighty issue that nags at you.
If your luggage is too weighty, you'll have to pay an extra fee at the airport. And if you've got a mind full of weighty problems, you won't be able to fall asleep on your long flight. Sometimes this adjective is also used to mean "influential," like the weighty influence your SAT scores might have on your chances of being accepted at a competitive college.