Teiki is a kind of fancy word that grown-ups use to talk about something called a "subscription."
Do you know what a subscription is? It's when you sign up for something to be delivered to you on a regular schedule, like getting a magazine in the mail every month.
In some places, people use the word teiki instead of saying "subscription." But it's not just for magazines or things like that, it could also be for things like public transportation, where you pay a certain amount of money to ride the bus or train for a certain amount of time.
Basically, teiki just means paying in advance for something that you will get regularly over a set period of time.