Most often when one hears the term Common-Law, it's in reference to marriage. If that's what you're looking for, go here. Otherwise, keep on reading.
First, let's start with the USPTO's definition:
"Common law rights arise from actual use of a mark. Generally, the first to either use a mark in commerce or file an intent to use application with the Patent and Trademark Office has the ultimate right to use and registration."
The above sentence really needs a caveat added to it: Yes, you can be 'first in line' if you file an Intent to Use trademark application PROVIDED that there are no other parties that can claim prior Trademark or Common-Law usage.
Also, Common-Law rights are restricted to the geographic area in which the mark is used.
Let's look at an example to illustrate this point:
In 2003, you started a web design company called Golden Fog Design & are based in the Bay Area of California. Your clientèle has been residents of the Bay Area strictly. This is where your Common-Law rights extend to, even if your web site that advertises your services is viewed worldwide.
Yesterday, a company called GoldenFog launched their web site advertising their web design services. They are based out of New York, NY and have been in business with that name since 2005. The company also filed for an Intent to Use trademark application with the US Patent & Trademark Office.
How does this work out? Now, of course, each situation varies from the next & I'm using rather crude examples to illustrate a point. So yadda yadda yadda, if this is happening to you, seek advice.
That being said, here's how it'd typically work out:
You'd still have your Common-Law rights to the Bay Area of California since you've been in business the longest. The NY company would have Common-Law rights to their city since they've been using it longest there. Now, the USPTO would also likely grant them the trademark since the USPTO will be blissfully unaware of your usage of the mark, UNLESS you contest the application.
However, even contesting it does not necessarily negate the Federal trademark. You may able to restrict the NY based company's Federal trademark registration from advertising & servicing the Bay Area of California.
Again, if you find yourself in this boat, contact a trademark specialist or a trademark attorney. Each situation is specific & the details will need to be hashed out.
Read more about Common-Law here.