The Internet Protocol, or IP for short, is a best effort protocol made for sending packets between computers without caring what types of networks are between them. IP is a best effort protocol since it does not make any guarantees that a packet will reach its destination. Every computer or node on an IP network is assigned address. A subset of these nodes are routers which know about relative locations of different addresses - a packet sent to a router will be passed along in the right direction. A node typically sends an IP packet by addressing it to its gateway, the local router, and lets the gateway send it out into the world. Hopefully, the packet will eventually get to the right node.
Besides providing addresses for routers to work with, IP also provides another protocol identifier so the receiving node can pass it to the right handler, another checksum in case of problems at the hardware layer and the ability to handle fragmentation (what happens when the hardware layer uses smaller packet sizes).
An IP packet traverses the network by being sent as the payload of a ``lower level'' protocol such as Ethernet. A machine with an IP packet has a few choices for what do with it. If the packet is at its destination, then the protocol field in the IP header will tell it what to do with the payload. If the packet's destination is mentionned in the routing tables, either specifically or in a block of addresses, then the machine knows what the next hop and which interface to use and can make the appropriate low-level packet to send. If the destination is not mentionned in the routing tables, there is generally a default route that is used.