In a simplified view, the public Internet can be viewed as a large interconnected network.  If Node A in the network is able to connect to Node B and vica versa there must exist a path between the two nodes.  This path undoubtedly passes through other nodes in the network (unless Nodes A and B are directly connected).  This path is virtual and may span over countries, continents, ISPs and even satellites.

Information between Node A and Node B are divided into packets that traverse the nodes between the origin and destination.  The packets contain information such as Host Address, Destination Address as well as the actual data.  When Node B receives the packet from A, the Host Address is still intact and therefore Node B is able to determine where the packet originated from.  Therefore, any website (Node) that you visit (send data to) on the internet is able to determine the source of the data (packet), you (Node.  Let us ignore NAT and other 1:N types of routing)

Tor allows Node A to anonymously send data to Node B without Node B being able to determine where the data originated from.  When a TOR client is started, a list of address (IPs) and other information concerning the entire TOR network is received.  Based on this data, the TOR client decides upon a random path in the network. Let the path be:  A->D1->D2->D3->B.  Therefore a packet of information will travel between these nodes in order to reach the destination B.  Node A first encrypts the data with the key of D3, followed by an encryption using D2 and lastly an encryption using the key of the next hop, D1.  Encrypting data with a key of Z means that only computer Z can decrypt this data.  The packet is then sent from Node A to Node D1.  Node D1 decrypts the ‘first layer’ of encryption and then forwards the packet to the next hop which is contained in the decrypted data.  This process continues from D1 to D2 and from D2 to D3.  When D3 receives the packet, the final encryption is removed and the original unencrypted packet is left.  Node D3 then sends the data to Node B, the destination.  (The original packet must be modified such that the Origin is now of Node D3).  Node B now believes that D3 (end node) is the source of the data.

Due to the encryption, no TOR node (D1,D2,D3) knows the complete path of the packet.  Each node only knows the immediate hop before and the immediate  hop after it. When Node B tries to send a packet back to Node A, he will first send it to Node D3 who will forward the data to D2.  The encryption happens in reverse this time.

Implications:

TOR is becoming more popular with groups requiring anonymous posting of data and information.  Journalists can use TOR to safely communicate with informants and even government employees can use TOR while abroad to hide affiliations with a particular country.  While TOR can be used for good things it can also be used for spam and other harmful activities.