The first is the layering of the different features. Which road is below and which is above? This can get complicated for large motorway junctions etc. where several bridges cross each other. In the easiest case the bridge gets the tag layer=1 while everything below doesn't have a layer tag (which means layer=0). Layers from -5 to 5 can be used. See Key:layer for details.

When a bridge goes over a waterway like a river, please tag the bridge=* with layer=1 as well. Do not add layer=-1 to the waterway, since this might cause issues with some renderers where the river is in a landuse=* or natural=* area. This also conforms to the common point of view, that everything at ground level should be the default layer of zero.

Also that part of the road, railway, etc. which is the actual bridge (compared to ramps etc.) gets the tag bridge=yes.

In most cases you'll have to break a way into three ways, one before the bridge, one for the bridge itself and one for the part after the bridge.

If the bridge ends in a junction, you'll need a small non-bridge way between bridge and junction (see image).

If the bridge has a name, you can use the name key: name=Waterloo Bridge.

When there are several bridges close together it sometimes is better not to tag them as bridges, but to tag the single road below as going through a tunnel.