Much like Outlook's filters, Gmail filters can be applied to any field in an email (to, from, subject and body). Google provides a complete rundown of gmail's advanced filters in its help section: http://mail.google.com/support/bin/answer.py?answer=7190
Not only can gmail's operators be used to filter email, but they can also be used to search email. Once you get a solid grasp of the filters available, you should be able to rapidly search and filter your way through your inbox with precision.
Now that you know about Gmail's published operators, its time to learn about some of the goodies that Google doesn't publish. Gmail offers a number of incredibly useful, but undocumented operators that you can use to search and filter your email. These filters can give power users superpowers.
Why its useful: Filter messages that you have been bcc'd on. If you've been bcc'd on a message, your email address won't appear in the to or bcc headers, so the deliveredto is the only way to filter those messages. For example, we bcc all of our automated outgoing mail to a plus-addressed gmail account and then filter and label those messages using the deliveredto field.
Operator: listid
Sample usage:
listid:djangousers.googlegroups.com
Why its useful: This filter is more reliable than the "list:" filter that gmail has documented as it provides discrete identification for lists that utilize this header.
Operator: replyto
Sample usage:
replyto:support@gap.com
Why its useful: Some newsletters vary their 'from' email address (they'll use addresses like 'santa@ecommercesite.com') while keeping a consistent reply-to address. The replyto: filter can help filter newsletters or emails that would otherwise be unfilterable.
Now that you know what the operators are, it is also important to walk though how you use them. You can search gmail using these filters simply by entering them into the search box in gmail. This is probably the most straightforward mechanism for applying your new found skills.
However, the second and more powerful approach to utilizing the new operators is to set up automatic filters. To get to the filter menu, click on the 'Settings' link in the top right of the page. Next click on the filters tab and then on the 'Create a New Filter' link. You'll notice that the list of fields include 'From', 'To', 'Subject', 'Has the words' and 'Doesn't have'. Since these operators are intended to access fields that aren't listed in Gmail's filter menu it would seem that you might be stuck. Fortunately, the 'Has the words' field functions just like the search box - you can enter your filter operators directly into this field.
Now that you are a power user, you need to use your new found power wisely. Try creating a few filters to label and archive certain newsletters (you'll always be able to find them again), or perhaps test your powers by writing search queries that return exactly the message you're looking for.
