To overflow is to go beyond filling something with a liquid, so that it gushes over the edges. During heavy rainstorms, rivers sometimes overflow their banks and flood the surrounding land.
Bathwater will overflow the tub if you don't turn the water off, and if you keep pouring coffee after your cup is full, it will also overflow. Literal overflowing involves liquid, but you can also use this verb to mean "fill a container" or "fill with feeling." So your arms can overflow with flowers, and your heart can overflow with happiness. The Old English root is oferfleow, "flood" or "inundate."