Think of a network as large room full of people all having conversations. Think of the internet as a collection of several of these rooms.
In this Allegory People = Computers, Rooms = Networks, Doormen = Routers.
Let's say you strike up a conversation about last night's episode of The Bachelor with your buddy over on the other side of the room. Anyone in the room can hear you and know what you are talking about, embarrassing as it may be. (Putting aside how a Switch works for a moment.) The fact of the matter is at any time someone in your room could be listening in on your conversation.
Now let's say you want to have a conversation with a banker about the balance in your account. Unfortunately the Banker is in another room 2 doors down. To accomplish this you need to talk to the Doorman of your room, and ask him to relay a message for you to the room next door, and that the Doorman next door to relay the message on yet again to the Doorman at the bankers room. Finally the Doorman of the bankers room shouts your message into that room so that your Banker hears your message and can respond, following the same method back again to you.
Unfortunately everyone in your room, the 3 Doormen along the way, and everyone in the Bankers room are now privy to your entire conversation. Is this something you are ok with? What if 1 person in one of the rooms is malicious? What if one of the Doormen is a shady character lacking in moral fiber?
The way to work around this to 'Talk in Code' with your Banker. Before you and your went in to your respective rooms, you agree upon a cypher to change your words. So rather than 'What's my Balance?' You would say 'green ideas sleep furiously', your banker would understand this and respond with 'tipsy trolly 12pm'. This sets you at ease because you know that that means your account has $2500 in it.
That 'Talking in Code' conversation is pretty much a VPN. People in your room (network) can hear your conversation. If you want to have a conversation with someone in another room (network) you need to rely on the Doormen (routers) to relay the message for you. If you don't want people to be able to decipher your conversation, you need to talk in code (vpn).
A VPN is a way for you to secure the conversation between yourself and a person in another room.
This allegory also works pretty well when trying to explain SSL.