after verb it means short action

verb + 一 + verb or verb + 一下 means an action happens very soon. for example, 让我想一想。or 让我想一下。 let me think. or let me see. it means let me think a little while. I need think but i just need a second for thinking. another example, 我能看一看(下)你的新手表吗？ can I see your new watch? it means I want to see your new watch but I just see it a little while---one second. but this usage is just for those verbs whose actions can be last. for example, think, see, listen, teach, etc, for these verbs: come, go, die, etc you cant use them like this: verb+一+verb or verb + 一下 normally, in any normal conversation, we always say like this: 让我想一想 我们一起去逛一逛街(逛街：go shopping) etc if you say 让我想，we can understand but it is really really weird.

一下 是很口语（informal）的说法，一般不用在正式(formal)用语中。比如：等我一下=等我（wait for me）

一下：express one kind of sentiment or mood , 快来看一下 : express so eager ,so insterested ,so surprised ,so excited .

一下= a short time,at once 1. Adv. the action occured or done in a short time ex. 这么多的生词，你一下就记住了。 2 a classifier of a verb.to indicate " Try to do or slightly mean" 去一下。 尝一下。