As part of the DOT code (G in the tyre marking above), there is a tyre manufacture date stamped on the sidewall. Take a look at yours - there will be a three- or four-digit code. This code denotes when the tyre was manufactured, and as a rule-of-thumb, you should never use tyres more than 6 years old. The rubber in tyres degrades over time, irrespective of whether the tyre is being used or not. When you get a tyre change, if you can, see if the tyre place will allow you to inspect the new tyres first. It's not uncommon for these shops to have stuff in stock which is more than 6 years old. The tyre might look brand new, but it will delaminate or have some other failure within weeks of being put on a vehicle.
Reading the code. The code is pretty simple. The three-digit code was used for tyres manufactured before 2000. So for example 1 7 6 means it was manufactured in the 17th week of 6th year of the decade. In this case it means 1986. For tyres manufactured in the 90's, the same code holds true but there is a little triangle after the DOT code. So for this example, a tyre manufactured in the 17th week of 1996 would have the code 176
After 2000, the code was switched to a 4-digit code. Same rules apply, so for example 3 0 0 3 means the tyre was manufactured in the 30th week of 2003.
http://www.carbibles.com/tyre_bible.html