Normally no CCC will allow you to balance transfer from one account with them to anotherwith them (i.e. Chase to Chase will not be approved) - I know for a fact Chase does not permit this. So I'll assume you mean you have a balance on a Citi card and you want to open a card with a different CCC and transfer your Citi balance over.
The maximum amount you can balance transfer, is going to be the Credit Limit of the new card you open. So if you have a Citi card with 10k debt, and you open a new card at 0% interest for 12 months but it's Credit Limit is 5k, then you can only move $5k minus 3-5% (depending on the balance transfer fee, some cards are free but its rare) over. Some places like USAA only permit 95% of the total credit limit to be used for balance transfers. Others, like NFCU, give you $0 bt fee and 0% for 12mo and let you completely max out the credit limit of your card with them.
How a balance transfer works is, you open a card, you give them the Citi cards account number, then they send Citi the amount you requested and they put a balance transfer transaction on your new card, plus any fees if applicable. It's pretty easy to do, just have to be patient because it can take anywhere from 2-3 days, to 2-3 weeks. USAA new cards have a 2 week waiting period before they will start the balance transfer - so be sure to ask whenever you're doing the balance transfer to see how long it will take. Best of luck!
Starting Score: 675