托(tuō) - to hold up on the palm, to support; to entrust, to ask someone a favour in Mandarin

The physical sense of 托 is holding something up from underneath on an open palm: a tray, a chin, a candle. That is different from 拿 (to take, to hold in the hand generally) and from 举 (to raise up high). Waiters 托 trays; you 拿 your keys.

The abstract sense grew straight out of the physical one. To 托 someone with a job is to place it in their hands: 托人办事, 托他带个话. This is where 拜托 (bàituō, please do me this favour) comes from, and it is the sense you will meet most often in conversation.

Watch the homophone. 托 (support, entrust) is not 拖 (tuō, to drag, to delay), and they differ only in a stroke or two. 拖时间 is stalling; 托时间 is nothing at all.