How much can I bargain down the cost of a used car?
Each car you find will answer this question differently. It's worth rereading Wha Who's excellent advice.
You need to figure out what is a fair or good or great price for each car you're looking at as a possible purchase. Some cars are drastically overpriced and have lots of negotiation built in. Rarely, you'll find a great deal that's fairly priced or an outright great deal.
I recently bought a GM certified used car (15 month-old Impala, 10,000 miles) that was priced at $16995, put on a special sale a few weeks later at $14995. I tried to get the price down more because I really wanted to spend $14000 max, although the sale price was very fair. I walked and they never called back. A couple weeks later, I tracked the car (online) as it moved to another GM dealer lot (a different town) owned by the same owner. I called and asked if they could knock some money off. They said probably not, but the owner was coming in later that day and they'd ask. Next day, I bought it for $14000 cash.
Other used cars I've found, even on dealer lots, were priced very fairly and I didn't haggle much at all because I didn't want to miss out. Some were in exceptional condition and on one, I got the dealer to replace the timing belt (due soon) and water pump and a couple of other items free of charge.
I wouldn't put a lot of stock in thinking "certified pre-owned" means the car is necessarily in better condition, but depending on the age and miles, the warranty that comes with "certified" is sometimes fantastic. I got the balance of 60 month / 100,000 mile drivetrain warranty and the bumper-to-bumper warranty was extended to 48 months / 48,000 miles. This keeps any possible buyer's remorse in check.
Whatever you do, I'd make sure the car is professionally checked (unless you're up to it) for any accident/collision repair. I don't buy those cars.
Do your homework. Look at several (many) cars to learn what a good deal looks like. Figure out what's fair or a good deal or a great deal and then buy or leave your phone number and walk away.
What model and model-year is this vehicle and how many miles on it ? What is the "certified" warranty that's included ?