What is the name of this metal bar with holes at the ends?
(so far, other answers don't show something I'd use on a 2x4, hence this answer)


  Two of these can go around a 2x4, then bolts run through the holes ... Is there a common application where these appear? 


"Straight Timber Connector" - for non-structural connections such as landscaping.



Smaller versions are usually called mending plates in my part of the world.



For structural applications involving vertical shear loads - like splicing joists - in buildings, you'd typically use something very different to that illustrated above. E.g. a "splice plate" or "nail plate".





As Isherwood noted, there are likely to be a variety of names for these types of connector.