You can use Oraclize to do exactly that. (It's an example of an Oracle - a mechanism for getting real world information into and out of a contract.)

There are some code examples here as well as the documentation of our Solidity API.

Oraclize is available both on mainnet and testnet so poking around with it should be easy, however if you need any support feel free to ask - we even have a gitter channel here.

As you can see fetching data from a website is as easy as using the oraclize_query function.

Our good behaviour is granted by the TLSNotary proof and can be easily checked with this web based client-side network monitor.

For example to fetch the ETHXBT price from the Kraken ticker: