Like with all "web applications" (a webmail service like Windows Live Hotmail is a web application), you can check your emails and your Hotmail account from anywhere you have an internet-connected computer. You never have to download anything, nor make sure that you are using the latest version of Hotmail: your web browser automatically downloads the latest files you need to use your account or send and receive email messages. But what about using your Hotmail email address on multiple computers at once?
The Windows Live team allows to have multiple people connected to the same Hotmail account at the same time, which means that you can easily have a Hotmail email address for your business, and allow several people to check for new emails and respond to emails from the same account.
There is nothing special to do in order to have a multi-user Hotmail email account: just create a Hotmail account to share among yourselves, and let each of you login from their own Windows, Mac, or Linux laptop or desktop computer. The only major hurdle to overcome when several people are using the same Hotmail account is simply to come up with a convention concerning mail to delete: you want to ensure that no email message is deleted in the assumption that someone else has already addressed it.
For convenience, you can even have several computers set to automatically login to Hotmail without any problem: when they load their inbox, these users will not have to go through the sign in screen (login form), except now and then to confirm that they are legitimate users, as is normal even for single users.
Tip: if you are using your Hotmail account with other people and everyone is allowed to respond to email messages, make sure that whenever you reply, you also blind carbon copy yourself (the shared email address) in the reply: this lets everyone know that an email has already been replied to.
The reverse is true as well: you can sign in to multiple Hotmail accounts on the same computer, simply by using different browsers or Windows/Mac usernames.