"I showed the textbook to my friend."
Te is used when you are asking for something and usually paired with kudasai for politeness. Like ドアをあけてください。doa wo akete kudasai. Please open (akete) the door. The dictionary form would be akeru. The same for misete (let me see); the stem for misete is miseru.
From what I've deduced from this course and the comments I remember:
te+imasu: continuing actions/actions in progress
te+kudasai: when asking to do something
Someone else can probably give the proper grammar terms for those forms and some other uses.
て is used as an action verb like -ing. Like when you are currently doing something, verbています. or if its a command, like verbてください 'please do verb', or in a list of doing different verbs, all the verbs before the last one are て. There might be a few other reasons to use them. But that's what I know off the top of my head.