Prepositions relate different items to each other. In English, many prepositions have numerous meanings depending on the context in which they are used. When a verb is followed by a preposition, it is often clear which one must be used, e.g. "rely on", "speak to". However, when a choice of preposition is possible, the meaning changes accordingly: "check for"; "check on"; "check over".
Prepositions indicate place or time (e.g. "Drive to London by 6pm"). Much like transitive verbs, prepositions need an object, and they are usually followed by a noun phrase ("be on time") or an "-ing" clause ("thanks for driving").
The technical term for a preposition and the noun phrase that goes with it is a 'prepositional phrase' - but your tutees only really need to know that if they're studying University English. For GCSE and A-level English, just make sure your tutee has a good grasp of how and when to use prepositions in their essays and creative writing.
Prepositions range from the simple ("to", "on", "in" etc) to the complex ("due to", "together with", "because of" etc) and can be strung together in prepositional phrases, as mentioned above.
Here are some examples of sentences containing prepositional phrases:
"The woman in the blue dress smiled at me."
"Dad put the chocolates on the table."
Prepositions can also combine with some verbs to create new meanings - these are called phrasal verbs. E.g. "He went through a hard time".