An alternate account (or "alt" for short) is an account used by a resident for something other than their usual activity or to do things in privacy (such as building or scripting). Alts are sometimes viewed in a negative regard as they have been used by some residents to abuse others anonymously and/or in order to avoid having their "main account" banned. However, alts have many legitimate uses as well such as using the account to role-play, build/script without being disturbed, hold land, aquire more stipend, and more.
Accounts that are on the same credit card could be considered alts of each other, although an alt usually refers to one person with multiple accounts rather than two different people on different accounts that are paid for by the same credit card (such as spouses or families).
It is impolite and against the TOS/CS to reveal an alternate account if the owner of the alt has not done so themselves. Accusing another resident of being an alt is also impolite.