For a small number of users accessing your file via a LAN or web publishing, you've already got what you need.
Open the database on one computer. Select Sharing | Filemaker Network...
Turn on sharing--this enables sharing from this computer to others.
In the "network access to file" section, select all users--this enables others to connect to this specific database file.
Now, from another computer on the network, launch FMP and select "Open Remote" from the file menu.
You should see your first computer listed along with its IP address. Select it and you'll see any filemaker files open on it appear in the second "pane" of the window.
Select a file and click open to start accessing the database.
In FMP 10, up to 9 users may simultaneously access your Filemaker hosted database. You must, however, have Filemaker installed on each of the client computers in order for them to access the shared database.
The other option is called "Instant Web Publishing" and it publishes your database to the web were others can access the db using their web browser. It's a lot more complicated to set up and not all features of FMP work with web publishing, but it avoids the necessity of each user needing their own copy of filemaker.
There are knowledge base articles and other threads here that deal with IWP and Hosted database you can research to learn more.