There are a whole lot of terms in your photos.
That vertical structure is referred to as a PIER. Another style of pier which as far as I know is a series of piles and a pier cap is referred more specifically as a BENT. Piers perform the job of supporting the bridge between abutments by carrying the vertical (and horizontal loads) to ground. The vertical element of the pier the arrow is pointed at I would classify as a PIER SHAFT or PIER COLUMN. The material being steel plate. The horizontal piece at the top attached to the leg, is referred to as the PIER CAP. It just is to support the girders and transfer their loads to the shafts. The shaft carry the load down to its footings, normally below ground, which are normally spread footings or piles.
Also of note in this photo are the GIRDERS. There are 4 of them These are the main longitudinal structural shapes supporting the bridge. Girders are normally either steel, concrete, or wood. They can also come in a variety of shapes, I, box, and trapezoidal being some of the more common ones. In this case you have steel plate I girder. Their job is to carry the vertical load of the deck to its supports (abutment or pier normally). This particular bridge is a bit of an odd ball as it has additional smaller longitudinal members running parallel with the girders. I would call these smaller members STRINGERS. They are transferring the vertical load from the deck to the Diaphragms.
Above the pier cap you will see a transverse steel plate member running between girders and it tappers as it extends out past the exterior girder. This member is referred to as DIAPHRAGM. The diaphragm performs several tasks. The main being to help redistribute the load between girders so the girders deflect more evenly, or behave as a group. The other job they do is help prevent twisting of the girders. Not that big of an issue on short straight bridges, but can be more of an issue on curved and longer spans. At the abutments and piers, the diaphragms also tend to be a bit beefier. This allows them to be used as jacking points. This allows the bridge to be lifted slightly to allow bearings to be replace and minor repairs if any being performed around the bearings. The diaphragms will also help transfer some of the horizontal loads the bridge deck may experience. Due to the stringers that are being supported by the diaphragms, one could argue that the diaphragms in this case are actually FLOORBEAMS. I would personally hesitate to call them that though as that is a term more often used with truss style bridge as opposed to a slab on girder bridge which is what I was assuming was shown in photo 1.
Beneath the girders you will notice some small steel members in an X pattern. This is commonly called CROSS BRACING These are normally L shaped sections, but can also be flat plate or rods to name a couple of other options. Their designed normally so one diagonal will be in tension and the other diagonal is in compression. The compression is assumed to overload the capacity of the diagonal and it will deflect/buckle out of shape and therefore not carry significant structural load. However the members will work in tension quite nicely. Since tension resistance is pretty much based on pure cross sectional area, it does not really matter if the member is a rod, L-section, plate or a cable in theory (never seen one as a cable though).
The portion outside of the outer most girder (exterior girder) is referred to as the CANTILEVER, bridge deck cantilever, deck cantilever, etc. It gets the cantilever name by the way the forces from it are transferred back to the exterior girder.
The arrow pointing up is pointing at a portion of what appears to me to be a cantilevered bridge deck. If you are referring more to the under side of the deck as opposed to the extension of the deck from the longitudinal support then the term I would use is SOFFIT. Soffit means the under side of the deck or ceiling.
The arrow pointed to the left I would simply call a wall. It appears very solid. I am not aware of a special term for it and will add that it is not common for a bridge as it adds significant weight. Bridge design tends to try and keep bridges light.
The arrow pointed to the right I would refer to as a BARRIER. There are 3 common forms of barriers. Parapet wall which looks very rectangular and is usually made of solid concrete. It can be topped with a pedestrian handrail if appropriate. Barrier wall which has beveled traffic face with a toe portion. Its similar to the parapet wall except its design is such that the car tire will impact first normally and it helps redirect the car slightly before the rest of significant impact occurs. The third type is an Open railing system. In this case you have an open railing system and more specifically a post and railing system
The left arrow is pointed at the end of a CROSS BEAM. This is one element that makes up a truss style bridge. Its job is to transfer horizontal load to the adjacent truss. It also helps improve the torsional stiffness of the bridge so the truss does not twist and collapse over long spans. Truss bridges where you have the tops of the truss connected and the traffic passes through the middle of the truss are referred to as Through Trusses. There is a shorter style of truss bridge where there are no transverse members attached at the top. It has a couple of names I am aware of: Pony Truss and Saddle Truss.
The right arrow is pointed at a member in a truss system call a VERTICAL. As pointed out by others, The vertical plan of a truss is made up of 4 main types of elements. Verticals which as the name suggests go up and down. Diagonals that go from top to bottom on some angle. Top chord which is the upper usually horizontal members. Bottom chord which is the lower usually horizontal member. Where the chords connect with with vertical or diagonals is usually referred to as a Node, though that comes up more for structural analysis purposes, and may just use connection in the field. Normally the top chord is in compression and the bottom chord is in tension.
Also of interest in this photo is that the transverse member instead of being an I beam or Channel, or Angle (L-section), is what I would call a built up member. Without taking too close a look I would guess it is a pair of Channels (C-Sections) back to back and stitched together with baton plates. This question has a more detailed description of this member make up.
Also note that small roughly 45 degree short member at the end of the transverse beam that connects to the vertical. This is commonly referred to as a knee brace. Its job is to help stiffen the connection of the transverse member to the vertical. This is what helps with the overall torsional stiffness of the bridge.
This is a DIAGONAL as discussed in an earlier photo. If this is the last member of the truss, meaning it's lower end is resting on a pier or an abutment, then you may also refer to this as the end diagonal. It will be the diagonal carry the largest axial load of all the bridge diagonals. As a result the end diagonal will usually be a larger member than interior diagonals.
This is an END DIAGONAL. See description of photo 4.
The lower arrow is simply a truss or frame being used as a pier to support the bridge. If the pier trusses were all interconnected, you could refer to this as trestle. Though trestles are more commonly made out of wood and more often than not rail bridge. There are definitely exceptions to this though.
The upper arrow looks like its some sort of wall. I would suspect non-structural and more like a debris fence. Really hard to say though with out getting different angles and closer look.
The black material looks like some sort of wrap used to either catch debris and keep it from falling on people and property below. If if was only on the truss members I might guess it would be netting to prevent birds from nesting and crapping on the bridge. Bird crap can be quite corrosive for steel. Oh I just thought of this, if that structure is slated for demolition, the black covering could be a blast net / mat intended to contain the shrapnel / flying debris caused by the use of explosives in explosive demolition.
The upper arrow seems to be pointed at some red pipe. I would assume this is steel pipe and it is painted red to make it stand out so people know its part of the fire suppression system as opposed to the plumbing for the roof drain. It is not a structural member.
The lower arrow is pointed at a beam. I would probably refer to this as a cross beam. I may also call it a brace as it is connecting the two columns.
The arrow on the left is pointed at what I would call storage tanks. If its oil or gasoline I would refer to them as fuel tanks. notice the wall around the tanks. This is a safety precaution. Those walls make a second tank or bath tub if you will. odd are the volume inside that external wall is close to 1.25-1.5 times the volume of the inner tank. The outer wall should contain the contents of the tank in the event of a failure/leak of the inner tank.
The arrow on the right pointed at the round metal structure I would call those structures hoppers or silos. This is really depended on what is being stored inside them. Dry contents like sand or grain I would definitely use one of these two terms.
The other arrow pointed at the rather square structure I am not sure what I would call it from this angle other than a building or structure.
For the most part, metal used in bridges in north america will all be structural steel. The strength of the steel has changed significantly over the years, but it is all steel to the best of my knowledge.