. . . It will not cost you anything if you have an unlimited internet connection (as I alreeady stated) or if you have a capped connection and do not go over your cap. Otherwise it will cost you to be a node for bitcoin.
Note that while your internet use might not cost you anything extra under those circumstances, you do have to leave your computer running and have to leave Bitcoin-Qt running to be a full node. You also have to store the entire blockchain on your hard-drive.
There are lightweight client options such as MultiBit, Electrum, and blockchain.info. These all allow you to use bitcoin with reduced system requirements, but if you use them then you provide less resources to the bitcoin network.
If there were a financial incentive for running a full node (like there is for mining), then more people would be more willing to run full nodes instead of lightweight clients. Given the existing incentives it seems likely that over time the network will move towards a situation where eventually it will be mostly mining pool operators, web hosted wallet operators, and merchant payment processors that maintain the blockchain and relay transactions. This centralization and lack of diversity in the network may allow transactions to relay faster (since they will have less nodes they need to relay through), but will provide consolidated targets for those who desire to interfere with the bitcoin network.
Certainly some individuals will continue to run full nodes out of their own desire to participate, but as bitcoin catches on more and more, the numbuer of transactions and the size of the blockchain will eventually reach the point where maintaining a full node will begin to add more noticeable costs, such as the need for additional or larger hard drives.