To "support" someone means that you help them. It can also mean that you will help them if they need help.
Some different kinds of support include:
- monetary support, which means that you will pay for someone to live or do business
- emotional support, which means that you listen to someone's problems and give helpful advice
- physical support, which simply means that you help to hold something up so that it doesn't fall down
When you're talking about one specific thing that someone does for you, you can say:
Thanks for supporting me.
But when you're talking about how someone has supported you repeatedly over time, you can use the phrase "give ___ support":
She's given me a lot of support over the years.
You can also use "give support" when you want to explain what kind of support it is - monetary, emotional, or physical.