Disclaimer: At the time of this writing, I am a shareholder with ASICMiner.
Also be aware that the Bitcoin stock market is, like Bitcoin, very immature and not regulated. The risks involved are far greater than for a traditional stock market.
Like @Steven Roose just wrote, you can buy Pass-Through (PT) shares through two separate stock exchanges, BTC-TC and Bitfunder. I personally prefer the former due to its easier interface, but either works.
However, know that these are PT shares only. A PT means that you buy shares with someone who holds real shares with ASICMiner. In practice, it is much the same, but both PTs will charge a 0.5% fee from your revenues, and you don't get voting rights, so keep that in mind.
For 'real' shares, ASICMiner do not sell those directly. They were sold initially at a now defunct stock exchange (GLBSE) which turned into a bit of a disaster and after that, friedcat, the person in charge of ASICMiner, decided he did not want his stocks traded on any exchange for fear of similar accidents.
That said, there are a lot of people that hold 'real' shares, and these are sometimes put on auction over on Bitcointalk.org (check https://bitcointalk.org/index.php?board=73.0). In case you wint one of these auctions, the seller will transfer the sold amount by sending a message to friedcat with your email address and your Bitcoin address for dividends. I suggest using one of the escrow service providers on that forum to ensure the transfer goes smoothly and safely. Several escrow providers will do this for free, but it is customary to tip a few bitcents at least for their service.
Dividends for PT shares are paid to your exhange account and directly held shares are paid directly from friedcat to your Bitcoin address on file.
Beyond that, ASICMiner does not have a web page per se, but they keep everyone informed several times a week here:
Like I said, be very careful about investing in Bitcoins and cryptocurrency stocks. There is no regulation, no recourse, and seldom any personally identifiable information.
And again, I am a current ASICMiner share holder, so keep that in mind as you read this.
EDIT: I've posted a more comprehensive and up-to-date answer here: