Pipe on the wall. Water keeps running
Looks like it could be a flow activated trap primer:


  A trap primer (or trap seal primer) is a plumbing device or valve that
  adds water to traps. The water seals in traps are needed to prevent
  sewer gases from entering buildings, but because this water is exposed
  to the air, it is subject to evaporation over time in infrequently
  used floor drains, leading to the release of sewer gas into the
  environment. The trap primer mitigates this problem by injecting
  water, either directly or indirectly, into the trap to maintain the
  water seal indefinitely. - Wikipedia




Example of one here and as well as the source of the above image (link is to a PDF, only picked this one as it was one of the first search results).

Look to see if you can find a make and/or model number on the 'Tee'. You may need to use a small mirror or camera to see the backside.

If it is now constantly flowing, there may be on obstruction inside the valve preventing it from functioning correctly. It could also be broken and in need of replacement. Depending on the valve, you may be able to just replace the internals.