"I showed the textbook to my friend."
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.
It's not a particle or anything like that, it's a verb form called simply the te-form, even though it often ends in で too (f.ex. よむ/よぶ->よんで).
Like you've noticed and still will notice on this course, it has many functions, but what it basically does is connect two verbs in one sentence. (The ください mentioned is actually a verb too.) I think you can google it if you're interested.