Often used in the context of road bike frames, a relaxed geometry bike allows for a more upright riding position and a more stable ride.
The easiest way to spot a relaxed geometry bike is the length of frame's headtube. This will be much longer in comparison to aggressive geometry bikes of the same frame size. This allows the rider to sit more upright, not having to bend over so much to reach the bars,
Harder to spot is that relaxed frames will have what's known as a "slack head angle" which means that the headtube slopes back further towards the rider than on more aggressive bikes. This brings the handlebars a bit closer to the rider and makes the wheelbase of the bike longer resulting in a more stable, less twitchy ride.
Touring bikes are a good example of bikes that always use a relaxed geometry because they are designed more for all-day rider comfort than for pure speed.