1.
to wait eagerly for; to look forward to impatiently
To wait impatiently for something one is looking forward to; literally 'to make one's neck long', craning to see whether it is coming. Almost always in the て-form before 待つ: 首を長くして待つ.
Literally 'to lengthen the neck', stretching to see whether what one is waiting for has appeared. Almost always 首を長くして待つ.