<em>What is the difference between...?</em>

"Savvy" means knowledgable in a practical, streetwise sense. Apparently it comes from a corruption of the French "Savez?" (Do you know?)

If you "get a meaning" or "get what someone is saying", then you understand it. This is one of the many common, colloquial uses of "get".  If some one is trying to tell you something but you don't know what he/she "is getting at", they are trying to suggest something (but not say it openly).

"To cotton on" means to eventually understand what is happening.

You can use www.thefreedictionary.com to check meanings in future. You'll find example sentences on that site.

It seems that all of them is meaning for how deep we know about something, but is there any differences among of them?

1."Savvy"

2. "Get at"

3."Understand"

1- I know how to

2 and 3 are comprehend something and can be used interchangeably slightly better.

"Cotton on"- I haven't heard this for what you mean. 

Examples:

I know a savvy shopper for CHristmas gifts that nevtrer spends more than 50.00.

I get what you are teaching.

I understand your confussion with the words.