You (and everyone else) need to pay tax on a quarterly basis. For people employed by companies, this is usually taken care of through withholding, so most people don't really the notice the need to make this right quarter-by-quarter. Now that you're self-employed, you'll need to handle this yourself through the estimated tax payments every quarter.
(It's possible, for example, to get a refund for being overpaid across the whole year and still have a penalty for being underpaid in a given quarter.)
Your employment earlier in the year doesn't really impact this at all, except to the extent that (a) you might "max" out for the year, as mentioned below in the last paragraph, or (b) you might already have enough withheld so that you don't actually need to make a payment because your prior overpayment covers (in full or in part) your liability for the current quarter.
The self-employment tax is basically that you now need to pay both the "employer" half of social security and Medicare taxes in addition to the "employee" half that would have been withheld from your paycheck when employed by someone else. The current total rate (both halves) is 12.4% for social security and 2.9% for Medicare. Just add those to whatever you estimate your regular income tax to be based on your total self-employment income and make your payment. When you file your Form 1040, you'll file Schedule C and break this out into components (income tax, SS, medicare), but for the estimated payment you only worry about the total.
If you make over certain amounts of money, the SS stops and the Medicare goes to a different rate, so if you think that might apply to you get the details from IRS publication 15.