Target Corp. was thrust into crisis mode Thursday after a massive theft of customer credit-card data threatened to dent the discounter's reputation and hurt sales during the final days of holiday shopping.

News of the data breach broke on Wednesday. On Thursday, after Target disclosed that 40 million credit- and debit-card accounts had been compromised in the breach, which ran from Nov. 27 until Dec. 15, anxious customers clogged the chain store's phone lines, jammed its credit-card website, and left angry posts on the corporate...