Being sat in the saddle is the one place you don't want to become uncomfortable. Even worse, the wrong saddle could lead to an injury from friction or pinching. There are a huge variety of saddles and as many different cyclists. As such it can be a hard notion to grasp that the saddle on the your new bike is unsuitable and may need replacing. Many riders stick with the original saddle provided but it may not be suitable for their riding position - aggressive or relaxed. The saddle may also not be suitable for particular uses - saddles for off-road use may not be the best option for road riding and vice versa.
Comfort is crucial but also subjective. It is highly recommended to invest in the right saddle for you and your adventures. Your sitting bones will take most of pressure from the saddle, and everybody's sitting bones are positioned differently. This also changes with different riders posture and their riding styles. You may find yourself constantly trying to correct your position; moving forwards or backwards. Or maybe the saddle is rubbing your inner leg or you find the nose too long or too narrow. These are areas for concern and you may need to consider changing your saddle.
Finishing touches are another detail to look out for. Seams, stitching or anything else that can may cause chaffing or discomfort should be considered.
The materials used in saddles will vary depending on discipline. Providing an off road rider with a stronger structure perhaps, with durable materials for rougher terrain whereas a road saddle may tend to be lighter - how you choose and what you need to consider is fundamentally the same.