For non-US readers, Wikipedia provides an explanation of what they do:
Borrowers visit a payday lending store and secure a small cash loan, with payment due in full at the borrower's next paycheck (usually a two week term). In the United States, finance charges on payday loans are typically in the range of 15 to 30 percent of the amount for the two-week period, which translates to rates ranging from 390 percent to 780 percent when expressed as an annual percentage rate (APR) The borrower writes a postdated check to the lender in the full amount of the loan plus fees. On the maturity date, the borrower is expected to return to the store to repay the loan in person. If the borrower doesn't repay the loan in person, the lender may process the check traditionally or through electronic withdrawal from the borrower's checking account.
If the account is short on funds to cover the check, the borrower may now face a bounced check fee from their bank in addition to the costs of the loan, and the loan may incur additional fees and/or an increased interest rate as a result of the failure to pay. For customers who cannot pay back the loan when due, members of the national trade association are required to offer an extended payment plan at no additional cost. In states like Washington, extended payment plans are required by state law.
Payday lenders require the borrower to bring one or more recent pay stubs to prove that they have a steady source of income. The borrower is also required to provide recent bank statements. Individual companies and franchises have their own underwriting criteria.
Defenders of the practice say that they provide a valuable service, as banks are usually disinclined to make such small loans, since it's too much trouble for too small a return. The high interest rates translate to relatively low absolute sums of money.
Critics of the practice compare payday lenders to loansharks. It's not a sustainable practice. In addition, the US Department of Defense has called payday lenders "predatory", and a Federal law limits interest rates on loans to military personnel to 36%. Payday lenders have previously targeted soldiers.
A number of other US States have Constitutional or legal limits on interest rates. The payday loan industry also exists in the UK and Canada. In the US and Canada, borrowers seeking small cash advances to cover their expenses would be better advised to join a credit union; credit unions often offer such services.