Weathering
One way in which salt comes to be in sea water is through a process called "weathering." Weathering happens when rain passes through rocks and soil, dissolving some of the minerals found within and carrying them into streams and rivers that eventually empty into the sea. Although this forms only a small amount of the salt in the oceans, it is part of the phenomenon. The level of salt is low enough that it cannot be tasted.
Evaporation
Another reason for the seas being salty is evaporation. Rivers and streams deposit their low levels of salt into the seas, but the water evaporates, leaving behind the mineral residues. As this process is repeated constantly and incessantly, the salt levels build up to the point that the oceans become extremely salty. This process is also evident in places such as the Great Salt Lake, where the salt levels are around 10 times higher than the sea because their is no outflow.
Hydrothermal Vents
Deep down in the oceans are what are known as hydrothermal vents. These are locations in which the water has penetrated the sub-sea crust of the Earth, and becomes heated. Once it is hot, it dissolves the minerals around it, including salt, and then flows back into the sea. This is a similar process to underwater volcanoes that erupt and deposit salt into the sea. It is thought that hydrothermal vents are responsible for a significant amount of the oceans' salinity.
Salt Levels
It is unlikely the seas will become saltier than at present. An equilibrium appears to have been reached in which new deposits of salt from the three processes described are counterbalanced by nature. This happens because dissolved salts are disappearing from the water at the same rate at which new salt is delivered. This happens when salt in sea water forms new mineral deposits on the ocean floor as fast as rivers and hydrothermal vents create new salt.