The density of a network is determined by its ratio of links to nodes. The higher the ratio, the denser the network.
Typically, the higher the density of a network, the more powerful its network effects are. The interconnectivity of links serve to reinforce and strengthen the connections between other nodes. If you're friends with someone, for instance, whose friends with all your other friends, the strength of your bond is likely to be stronger than it would be in isolation.
Density is typically distributed unevenly within a network. Certain areas within a network can have much higher density than other areas of a network (which is what leads to clustering, a phenomenon described in greater detail below).
In building products, it's advisable to pay attention to how nodes form connections with each other so you can design your product to promote higher network density. Look for the "white-hot center" of your network - the densest, highest activity part -and focus the product features and language on activating other users to behave more like that group. Their activity will attract other nodes who will be inspired by the activity of the "white-hot" group, and it will radiate outward from there much faster than you might think.