How do I find a node knowing its URL?
The easiest way is almost always to visit the page you want to change, and hit the "Edit" button among the tabs.

However, sometimes this may not be practical for whatever reason. SQL-queries below:

D6

SELECT src FROM url_alias WHERE dst = 'home-care/vids';

D7

SELECT source FROM url_alias WHERE alias = 'home-care/vids';

You may in both version want to add AND language = if your site is multilingual.