Bridges are a piece of engineering which take months and sometimes years to build with many people being involved in the building process. Here's an overview:
Determine what type of bridge you need for the situation. The common types are arch, suspension, truss, cable and beam types. If you're crossing a short span, you probably only need to build a beam or a small truss bridge. If you're crossing a deep ravine, an arch bridge might be the best choice. A bridge to span an ocean bay requires a very expensive suspension bridge.
Identify an ideal place to build the bridge. A lot of legwork goes into determining the ideal site to build anything, and when you're talking about something as expensive as a bridge, engineers are even more careful. The engineering team will send out surveyors, people who use electronic measuring devices to make a map of the general area. From this map, they will make calculations and determine the length and placement of the bridge.
Design the bridge. The engineer will take into account the potential traffic load of the bridge and use steel design manuals, computer aided design programs to aid the bridge design process.
Once the design has been finalized and approved, the building can begin. This will involve excavation or filling the banks on either side, stabilizing the ground where the supports will be placed and, depending on what type of bridge is going to be built, pouring concrete.
Some bridges are built on site and some are put together in pieces small enough to take to the location and then assembled. Using the anchors points placed in the concrete, the steel is put together with huge cranes with nuts and bolts and welding.
Once the main structure is in place the surface can be put in place, this is normally built on steel cross members attached to the main supports. Concrete is then poured section by section until the surface is built. It is then a matter of either attaching rails if its a train bridge or adding a further surface like tarmac to make the final service to make a road.
You then get some important person to come open the bridge, ha-ha :)