See this question regarding the relationship between a HDHP (High Deductible Health Plan) and an HSA (Health Savings Account). In brief, to qualify for an HSA you must have a HDHP:
HDHPs are plans with a minimum deductible of $1,200 for self-only coverage and $2,400 for self-and-family coverage. The maximum amount out-of-pocket limit for HDHPs is $5,950 for self-only coverage and $11,900 for self-and-family coverage.
As mentioned by Stainsor, your insurance can either come from your employer, or it can be an individually purchased plan. The HSA can be bundled as part of a package with the insurance, or it can be an account you set up separately.
Contributions you make to the HSA are tax deductible. You'll report the amount you contributed when you file your taxes the following year. E.g. in April 2012 you'll report (and deduct) the amount of HSA contributions you made for tax year 2011.
I'm not sure what kind of trouble you'll get into if you have an HSA without having a qualified HDHP.
To answer the main part of your question:
Different HSAs may have slightly different features, but I've typically seen them provide the following ways to withdraw funds:
Via a debit card issued with the account. You can use the debit card to pay for things like drugs at the pharmacy, or at a doctors' office that requires payment at the time of service.
Via online bill pay. You can use this to pay bills from hospitals, doctors' offices, or other healthcare service providers that send you bills.
Via paper checks. For doctors' offices that require payment at time of service but don't accept plastic. (Or if you prefer not to use online bill pay.)
Via withdrawal at a teller window or ATM. You can use this to "reimburse yourself" for healthcare expenses that you paid out of pocket.
The issue of documenting legitimate expenses and/or qualifying for the account with an HDHP is between you and the IRS. The bank at which your HSA is kept doesn't really care whether you comply with the tax laws.