To beguile is to entertain and convince by flattery. Someone who beguiles is tricky and often charming, like when a smooth-talking friend tries to beguile you into giving them your allowance.
The word beguile is from Middle English, and while guile involves deceit, to beguile means to spend time pleasantly, even if it involves trickery. If someone beguiles you for an afternoon, you were amused, entertained, or otherwise pleased. The word can also have a darker tone, as when the person who beguiled you for the afternoon was actually trying to steal your car. Con artists and cat videos tend to beguile people.